(CNN) -- Henry "Hank" Skinner came within 45 minutes of lethal injection in March 2010 before the U.S. Supreme Court stepped in -- and ultimately handed him a legal reprieve.
But as another execution date looms, pressure is mounting on Texas officials to grant Skinner's long-standing request to have crime-scene DNA tested before it's too late.
"Executing Mr. Skinner without testing all the relevant evidence would suggest official indifference to the possibility of error in this case and needlessly undermine public confidence in Texas's criminal justice system," said a letter from former Texas Gov. Mark White and 16 other current and former Texas lawmakers, prosecutors and judges, dated October 27 and sent to Gov. Rick Perry, Attorney General Greg Abbott and Gray County District Attorney Lynn Switzer.
Skinner, 49, was convicted of the New Year's Eve 1993 killings of his 41-year-old live-in girlfriend, Twila Busby, and her two adult sons in the Texas Panhandle town of Pampa. He is set to die Wednesday.
He has strongly denied any involvement in the crime and claims that the DNA analysis of untested evidence will prove his innocence and help determine the real killer.
"All the district attorney has got to do is turn over the evidence and test it, and let the chips fall where they may," Skinner told CNN's Kate Bolduan in 2009 in an interview from death row at the Polunsky Correctional Institution in Livingston, Texas. "If I'm innocent, I go home. If I'm guilty, I die. What's so hard about that?"
On Thursday, the Texas trial court denied Skinner's latest motion asking to have the crime scene items tested.
"We are deeply disappointed that the trial court has denied Mr. Skinner's request for DNA testing," Rob Owen, Skinner's attorney, said in a statement. "Unfortunately, the trial court's order offers no explanation for its conclusion that DNA testing is not called for in this case."
The case will now go to the Texas Court of Criminal Appeals, Owen said, adding that he is confident the court will give the case "deliberate consideration. ... We are confident that upon such careful review, the court will conclude that DNA testing is necessary in this case to ensure the reliability of the verdict.
"But for now, the Court of Criminal Appeals must stop the scheduled November 9 execution rather than allow itself to be rushed to a hasty and ill-considered decision," Owen said. "The stakes in this case are too high to allow Mr. Skinner to be executed before he has a fair chance to make his case that the trial court made a grave mistake in denying his request for DNA testing."
The Court of Criminal Appeals is Texas' highest criminal court.
Owen said he is asking the Court of Criminal Appeals to ask the trial court to provide more information on why it made its ruling. A federal case in which Skinner is also seeking the testing has been put on hold pending the outcome of the state court proceedings, he said. If the Court of Criminal Appeals denies Skinner's motion, the federal court case will proceed.
Texas state attorneys argued at a recent court hearing that the testing should not be conducted because there was not a reasonable probability the trial jury would have found Skinner innocent if the testing had been done for his trial, Owen said.
Switzer's office refused comment on the case to CNN on Thursday. Perry's office referred questions to the attorney general's office, which also denied comment.
More than 120,000 people have signed an online petition calling for the testing.
In May 2010, after halting Skinner's execution in March of that year, the Supreme Court said on a 6-3 vote that Skinner has a basic civil right to press for DNA analysis, giving him another legal avenue to push his claims of innocence.
At the time, Justice Ruth Bader Ginsburg noted that Skinner's victory was limited and that the ruling only gave Skinner access to the DNA evidence, which might point to his innocence, point to his guilt or be deemed inconclusive.
However, Texas prosecutors have fought the testing, claiming that Skinner is not entitled to testing of evidence not analyzed before his 1995 trial. Forensic evidence gathered at the scene, as well as witness statements, points to Skinner as the killer, they maintain.
Skinner admits being at the crime scene when Busby and her sons died but said he was passed out on the couch from a combination of vodka and codeine and was too intoxicated to have committed the murders.
A female friend of Skinner's who lived four blocks away testified at his trial that he walked to her mobile home and told her that he may have kicked Twila Busby to death, although evidence did not show she had been kicked. The neighbor has since recanted parts of her testimony.
Authorities followed a blood trail from the crime scene to the female friend's home and found Skinner in the closet, authorities said. He was "wearing heavily blood-stained jeans and socks and bearing a gash on the palm of his right hand," according to the Texas attorney general's summary of the case.
Authorities said cuts on Skinner's hand came from the knife used to stab the men. Skinner claimed he cut it on glass. Some DNA testing was done, which implicated Skinner, but not on the items he now wants tested.
In addition to Twila Busby, also found stabbed to death were her sons Elwin "Scooter" Caler, 22, and Randy Busby, 20, both of whom were developmentally disabled.
"The victims' injuries show that whoever murdered them must have possessed considerable strength, balance and coordination," Owen wrote in the Supreme Court filing. Twila Busby was manually strangled so forcefully that her larynx and the hyoid bone in her throat were broken. She then was struck with an ax or pick handle 14 times, hard enough to drive fragments of her "unusually thick skull" into her brain, the court documents said.
"While attacking Ms. Busby, the perpetrator had to contend with the presence of her six-foot-six-inch, 225-pound son, Elwin Caler, who blood spatter analysis showed was in the immediate vicinity of his mother as she was being beaten," the court filing said.
"Somehow, the murderer was able to change weapons and stab Caler several times before he could fend off the attack or flee."
Among the items Skinner wants tested, according to court documents, are vaginal swabs from Busby contained in a rape kit; clippings from her fingernails; two knives, one found on Busby's front porch and the second found in a plastic bag in the living room; a dish towel; and blood and hairs from a jacket found next to Busby's body.
Skinner's attorneys have in the past presented what they call compelling evidence pointing to another man who may have committed the murders. Evidence presented at trial suggested that Busby's uncle, Robert Donnell -- who is now dead -- could have been the killer. At a New Year's Eve party she attended for a short time on the last night of her life, Donnell stalked her, making crude sexual remarks, according to trial testimony. A friend who drove her home from the party testified that she was "fidgety and worried" and that Donnell was no longer at the party when the friend returned to the celebration.
Sandrine Ageorges-Skinner, a Frenchwoman who married Skinner in 2008, told CNN's "Larry King Live" in March 2010 that she began writing to him in 1996, and they began visiting in 2000. Ageorges-Skinner has been an outspoken anti-death penalty advocate and a key ally in her husband's struggle.
"I'm convinced of his innocence ... (because) there is scientific forensic evidence to prove that he was not even in a state to stand up at the time of the crime, let alone murder three people that he loved," she told "Larry King Live." "There is absolutely no motive."
Ageorges-Skinner said Thursday that she was not surprised by the trial court's ruling.
"What can you expect from a sentencing judge who has signed four death warrants, denied two previous DNA motions and denied all (Skinner's) appeals?" she wrote in an e-mail to CNN. "This is not justice. When sentencing judges are expected to review their own work and decisions, one cannot expect a fair and impartial decision."
In an interview aired Thursday by France Info, a news radio channel, Ageorges-Skinner said the last few years have been "with the stress and anguish, very intense ... with moments of hope, since he had a new execution date. ... Also moments of laughter, moments of relaxation, because life is too short and you can't forget to live."
A documentary is set to air on the French television network Canal Plus on Tuesday, she told France Info. It was filmed last year, when a camera crew followed her in the days before Skinner was set to die.
"I try not to look at the calendar, even though every morning when I wake up, I know how many days are left," Ageorges-Skinner said at one point, according to excerpts of the documentary posted on Canal Plus' website. At another point, she reads a letter from Skinner telling her he does not want her to witness his execution because "I want you to remember me as I was in life, laughing, joking and talking."
Ageorges-Skinner told France Info that she has been barred from visiting her husband by prison administrators "because we both are exposing problems of incarceration."
Michelle Lyons, spokeswoman for the Texas Department of Criminal Justice, said Ageorges-Skinner was banned from her husband's visitation list a couple of years ago because of "security issues." Skinner was found in possession of a cell phone, she said, and his wife was taken off the list after some "unauthorized money exchanges around the same time."
Ageorges-Skinner will be allowed to visit her husband as his execution date approaches, Lyons said.
Ageorges-Skinner, however, said Sunday that she was banned six months before the cell phone incident. She said she was accused of facilitating trafficking between Skinner and another inmate by sending money to a prison account.
An investigation found no proof of trafficking or violations, she said, and the funds were credited to the inmate's prison account, but the ban stands. She said she appealed, but was rejected.
Texas has executed more prisoners than any state since 1976, when the high court allowed executions to resume.
Questions have swirled in Texas regarding the 2004 execution of Cameron Todd Willingham for a fire that killed his three daughters and claims that he was not guilty.
In 2009, Perry issued a posthumous pardon for Timothy Cole, who was serving a 25-year sentence for aggravated sexual assault when he died in prison from an asthma attack. After his death, DNA tests established his innocence, and another man confessed to the crime.
"We believe that the death penalty is an appropriate punishment for certain crimes, and we understand that the DNA testing might well show that Mr. Skinner is deserving of that punishment," said the letter from lawmakers to the Texas officials. "But we are also steadfast in our belief that when it comes to the ultimate penalty, we must to everything in our power to ensure certainty before taking the irreversible step of carrying out an execution."
The letter notes October's exoneration of Michael Morton after 25 years of wrongful imprisonment. That case "highlights why state officials should consent to DNA testing when untested evidence is available," the letter said.
"There is simply no justifiable reason why Texas continues to waste taxpayer dollars in its decade-long fight to prevent scientific testing in Mr. Skinner's case."
The Texas Criminal Court of Appeals initially determined that the evidence was not tested at Skinner's trial because of the strategy of his trial attorneys -- which the court determined did not meet the no-fault provision of Texas law, which then stated that inmates could access post-conviction testing only if the testing was not done through no fault of the defendant. That "no fault provision" was stripped from state statutes in June.
Busby's family has also pressed state officials to do the forensic testing, saying it would end the years-long delay while Skinner has pressed his legal claims. The family, including Busby's surviving daughter, believes Skinner is guilty.
CNN's Alanne Orjoux contributed to this report.