For a year and half, Mueller's family in this idyllic town has lived with the knowledge that the Islamic militant group ISIS is holding the 26-year-old American aid worker.
Publicly, they kept silent about her plight, with her captors threatening to execute her if the family spoke out.
But on Friday, ISIS claimed that Mueller had been killed in a building that was hit during a Jordanian airstrike on Raqqa, the militants' de facto capital in Syria.
"Friday was a dark day," said Todd Geiler, a longtime friend of the family. "Punched a hole through you. A big hole."
Still holding out hope
But ISIS offered no proof to back up its claim, other than an image of a building in rubble.
The Muellers are holding out hope that she's still alive, imploring the militant group to contact them.
"We have sent you a private message and ask that you respond to us privately," her parents, Carl and Marsha Mueller, said in a statement Friday.
"You told us that you treated Kayla as your guest," they said. "As your guest her safety and well-being remains your responsibility."
'A fluid situation'
Jordan is openly disputing ISIS' claim about Mueller's fate.
"What we know about this terrorist organization is that they are liars when it comes to these things," said Jordanian government spokesman Mohammed al-Momani.
He questioned the extremist group's ability to identify Jordanian warplanes flying at high altitudes.
But in Prescott, Mueller's family are staying out of the claims and counterclaims of what happened.
"At this time, they just want contact through the original channels," said Geiler.
"This is such a fluid situation, going hour to hour," he told CNN.
Questions over ISIS claims
Some observers have questioned why no militants died in the collapsed building, and why, if the building had just been struck, there was no smoke from smoldering debris.
But others say they fear ISIS may have killed her previously and been waiting for an opportunity to announce the death.
Jordan says it believes the militants killed one of its fighter pilots weeks before they released a video of him being burned to death.
ISIS has made a big show of its barbaric killings of international hostages, including three Americans, two Britons and two Japanese citizens.
The militants had threatened to kill Mueller last summer. They set a deadline of August 13, for a ransom of nearly $7 million to be paid to prevent her execution, according to a source close to the family.
U.S. officials are investigating the theory that she may have died weeks or months ago and that ISIS was possibly ashamed to announce that 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."
Commitment to international causes
Like some of the other ISIS hostages, Mueller went to the Middle East to help the flood of refugees affected by Syria's grinding civil war.
Since her teenage years, she had thrown herself into international causes.
In high school, her local paper showed her marching through town as part of the Save Darfur coalition, lobbying members of Congress and staging silent protests against the genocide in western Sudan.
As a student at Northern Arizona University, Mueller was president of a group called Stand, a student-led movement to end mass atrocities.
After graduating, she joined aid agencies that took her to India, Israel and the Palestinian territories. She came home briefly in 2011, volunteering at a women's shelter and an AIDS clinic.
'A living hell'
But she couldn't ignore the unfolding crisis in Syria.
"I am in solidarity with the Syrian people. I reject the brutality and killing that the Syrian authorities are committing against the Syrian people," she said in a video posted online.
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.
Her family heard nothing until 10 months later, when ISIS got in touch with its ransom demand.
"You have no control. You have to abide by the rules," Geiler said of the situation. "It was a living hell, and it has been a living hell for the family, and it is today."