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NEW: State news: UAE putting F-16 squadron in Jordan to reaffirm solidarity

Kayla Mueller's parents ask her captors to contact them privately

U.S. officials have no evidence she has been killed

Amman, Jordan CNN —  

The parents of Kayla Mueller are imploring ISIS to contact them. They believe – or at least hope – that their daughter may still be alive.

“We have sent you a private message and ask that you respond to us privately,” Carl and Marsha Mueller said in a statement.

The couple is not giving interviews to the media as ISIS warned them not to.

CNN had previously not published her name at the request of her relatives, who have now publicly identified her.

The terror group captured Mueller, an Arizona native, in 2013. It said Friday that a Jordanian airstrike that pounded their de facto capital of Raqqa in Syria also killed her.

Her parents, though, are holding out hope “that Kayla is still alive.”

“You told us that you treated Kayla as your guest,” they said. “As your guest her safety and well-being remains your responsibility.”

‘Liars,’ no proof

Jordanian officials challenged the veracity of the claim, with Interior Minister Hussein Majali calling it nothing more than a publicity stunt.

“What we know about this terrorist organization is that they are liars when it comes to these things,” said spokesman Mohammed al-Momani. He cast doubt on the extremists’ ability to identify Jordanian warplanes flying at high altitudes. Many are U.S. made F-16s.

Officials in Amman have been adamant about their intent to go after ISIS ever since Tuesday, when the terror group released a video showing its members burning Jordanian pilot Lt. Moath al-Kasasbeh to death.

They’ve vowed not to give up the fight, even after ISIS’ latest claim about Mueller.

On Saturday, they proved it.

According to state TV, Jordanian fighter jets conducted more airstrikes targeting ISIS position before returning safely to their bases.

Anti-ISIS activists reported that coalition strikes pounding targets around the Islamist extremist group’s de facto capital of Raqqa. Pro-ISIS reporters said “heavy airstrikes” hit more than a dozen areas.

Kurds say ISIS destroyed key Iraq bridge

Jordan isn’t going after ISIS alone. It’s part of the U.S.-led coalition also hitting ISIS targets in Syria and Iraq, helping local forces on the ground. And the United Arab Emirates will station an F-16 squadron “in fraternal Jordan” in part to reaffirm “the UAE’s unwavering and constant solidarity with Jordan,” according to the Arab nation’s state-run WAN news agency.

Questions, suspicions

ISIS has shown no proof the 26-year-old was killed, other than a building laid to rubble and the claim that her body was inside.

Experts have questioned why no militants died in the collapsed building, if they were holding Mueller there, or why – if the building had just been crushed – there was no smoke from smoldering debris.

Skeptics fear ISIS may have long killed her and just waited for an opportunity to officially announce the death – a suspicion that has been floated before when ISIS uploaded slickly produced, odious videos of other victims’ killings.

Summer death threat

Jordanian officials said they have reason to believe al-Kasasbeh was actually killed in early January, and that ISIS lied about him still being alive while trying to negotiate advantages in exchange for him.

The group has a penchant for executing, not freeing.

ISIS had implied they might kill Mueller last summer. They set a deadline of August 13, 2014, for a 5 million Euro ransom to be paid to prevent her murder, according to a source close to the family.

U.S. officials are investigating on the theory that she may have died weeks or months ago, but that ISIS was possibly ashamed to announce they had killed a woman.

But retired U.S. Army Lt. Col. James Reese says it’s plausible a hostage may have been inadvertently killed in Syria.

“We all know that intelligence … in Syria is not as good as we have in Iraq now,” Reese said. “So it is plausible, and it could be unfortunate. But sometimes … that happens in these wars,” he said.

Faith and compassion

Mueller landed in Syria following her faith, empathy for human suffering and wanderlust.

In her home state, she worked in an AIDS clinic and volunteered at a woman’s shelter. After graduating from Northern Arizona University, the Prescott native traveled to India and Israel.

“I find God in the suffering eyes reflected in mine. If this is how you are revealed to me … this is how I will forever seek you,” she said, according to her family.

In 2012, Mueller went to Syria with the Danish Refugee Council and Support to Life humanitarian agency.

She fell into the hands of hostage takers in August 2013 in Aleppo, Syria, her family said, after leaving a Doctors Without Borders hospital.

Litany of atrocities

The list of ISIS atrocities is long and growing. It extends far beyond the glitzy video celebrations of hostage murders.

The terrorists have wiped out villages of people whose religion they reject, summarily executing hundreds, kidnapping scores of women, then raping them and passing them into slavery.

They have stooped to selling children, a U.N. report this week claimed.

They aim to instill fear in people’s hearts, as they aspire to virtually take over the world with a caliphate, but their barbarism is having the opposite effect as well, unleashing disgust and wrath against them.

Putting al-Kasasbeh in a cage and setting him on fire has fueled resolve among Jordanians to stop the group’s smudging of their religion by killing in its name.

“This is definitely not the U.S.’s war,” said Majali, the nation’s Interior Minister. “This is definitely our war.”

CNN’s Jomana Karadsheh reported from Jordan and Ben Brumfield wrote this report from Atlanta. CNN’s Nic Robertson, Evan Perez, Greg Botelho, Steve Almasy, Salma Abdelaziz, Ali Younes and Barbara Starr contributed to this report.