Prescott , Arizona ( CNN ) The days pass one after the other in Prescott , Arizona , without bringing any real answer about Kayla Mueller . 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 . '' 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 . '' 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 . 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 . '' 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 . But she could n'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 . '' CNN 's Kyung Lah contributed from Prescott , and Jethro Mullen wrote from Hong Kong . CNN 's Ben Brumfield and Jomana Karadsheh contributed to this report .
`` It has been a living hell , '' a longtime friend of the family tells ISIS . CNN claims Jethro Mullen , an Jordanian woman , was killed in a American airstrike . Her family asks the militants , who previously demanded a ransom , to get in touch .