Kansas City, Missouri (CNN) -- The family of a missing Missouri baby said Thursday that it will not speak publicly until next week, making the announcement one day after Kansas City police said the girl's parents had declined to be interviewed separately by authorities.
The family, through its attorney, had scheduled a press conference for Thursday. But that event, and all others involving the media, were "being postponed until next week," the family said in a statement.
"The last few weeks have been exhausting to everyone working on behalf of the Irwin family, it has exhausted Lisa's parents and her friends and family," the family said. "Therefore, the consensus is we all need a rest."
Eleven-month-old Lisa Irwin was reported missing about 4 a.m. October 4, after her father Jeremy Irwin, arrived home from work to find the door unlocked, the lights on and a window tampered with. Her mother, Deborah Bradley, said she last saw Lisa at 6:40 the night before.
Authorities conducted a two-hour search Thursday for clues, including looking in and around Chaumiere Lake in Chaumiere Woods Park located in north Kansas City, about two miles from the baby's house. Two cadaver dogs capable of picking up scents in the water were part of the effort.
But there was no indication that authorities detected anything new in the search.
No one has been named as a suspect in the case. The girl's parents have been questioned by, and sometimes been at odds with, those investigating Lisa's disappearance.
That includes this week, after Kansas City police Capt. Steve Young said that investigators had planned to conduct interviews with each parent separately. But he said a day later, on Wednesday, that the couple had declined.
Young said that he did not dispute reports that the family had cooperated and answered questions previously. But he added that police department detectives still had unanswered questions.
The parents' attorney, Cyndy Short, explained that her clients are not opposed to separate interviews, insisting they have been cooperative and have previously been queried by police, both apart and together.
But she said that they will not agree to what police requested -- an unrestricted interview, with no attorneys present. The pair don't mind being interviewed separately as long as the detectives are fair, open-minded and non-accusatory, according to Short.
Meanwhile, Lisa's half-brothers -- who were in the family's home the night she disappeared -- will be re-interviewed by authorities Friday, Kansas City police said.
The boys will be interviewed by a "child services specialist trained to interview kids," Young said. The interview will be nonconfrontational, according to Young, and a police officer won't be in the room.
"Not an interrogation," the police captain said. "They are kids, after all."
The boys are 8 and 6, according to Short. Young said they were previously interviewed just after Lisa disappeared, both for under an hour.
Details were being worked out for a re-interview of the boys, Short said.
Police also said they are seeking DNA tests on the boys in order to eliminate them from all the DNA found in the home during a police search.
A cadaver dog searching the family's Kansas City home indicated a positive "hit" for the scent of a body, according to documents released last Friday.
On that information, Short said that law enforcement typically begins such investigations inside the home, then if warranted the probe moves outside the family.
"We don't know exactly what happened here, so we need to keep broadening the investigation," she said.
Between 30 and 50 investigators, including police and FBI personnel, were involved in the search as of Tuesday, the Kansas City Police Department said. Some 900 tips then had been received by investigators, the department said.
Short said that Lisa's parents don't know what happened to their daughter. She said that the couple has provided a wealth of information to authorities, including the names of people who have had access to their home.
"They don't know. I mean, someone came into their home in the middle of the night and took their beloved child away from them ... They are stumped," the attorney said.
CNN's Jim Spellman and Bill Kirkos contributed to this report.