PRESCOTT, Arizona — The parents of a 26-year-old American who Islamic State extremists say was killed in an airstrike in Syria said in a statement addressed to group leaders that the claim of their daughter’s death concerned them but they were still hopeful she was alive.

The Islamic State said Friday that Kayla Jean Mueller died in a Jordanian airstrike, but the government of Jordan dismissed the statement as propaganda. The U.S. said it had not seen any evidence to corroborate the report.

“Their nerves are absolutely frayed,” family friend Todd Geiler said Saturday after leaving the Mueller home. Mueller is the only known remaining U.S. hostage held by the Islamic State.

If the death is confirmed, she would be the fourth American to die while being held by Islamic State militants.

Journalists James Foley and Steven Sotloff and aid worker Peter Kassig were beheaded by the group.

“You told us that you treated Kayla as your guest, as your guest her safety and well-being remains your responsibility,” Mueller’s family said in a statement released Friday.

The road leading to her family’s home in Prescott remained blocked by authorities Saturday. FBI agents have been with the family and remain at their home, Yavapai County Sheriff’s spokesman Dwight D’Evelyn said.

Mueller is an aid worker who previously volunteered with organizations in India, Israel and the Palestinian territories.

“The common thread of Kayla’s life has been her quiet leadership and strong desire to serve others,” Mueller’s family said.

Her identity had not been disclosed until now out of fears for her safety. Her family said she was taken hostage by the Islamic State on Aug. 4, 2013, while leaving a hospital in Syria.

“The secrecy issue was at the demand of her captors,” Geiler said. “There were just a few of us around town who knew, for lack of better words, the living hell the family was going through.”

He said the family planned to release another statement sometime Saturday after meeting with U.S. government officials. He offered no further details.