"It is in their benefit as well and the benefit of all us," Yassin AlRawashdeh, a former diplomat, told CNN on Friday.
"Otherwise, they will get the anger of the whole people of Jordan and the anger of the whole world," he said.
ISIS had pushed Jordan to bring a convicted terrorist, Sajida al-Rishawi, to the Turkish border by Thursday evening. The Jordanians didn't, so governments and relatives were still waiting Friday to see whether ISIS followed through on its threat to kill al-Kassasbeh, followed by the beheading of a Japanese journalist, Kenji Goto.
Another of the pilot's uncles, Fahed Al-Kasasbeh, said the family hasn't received any information from official sources. And there has been no proof of life from ISIS.
"But we hope he is still alive," AlRawashdeh said.
Al-Kassasbeh's father, Safi, made a similar appeal for his son's release Thursday, after the sunset deadline.
Jordanian officials have said they are willing to swap al-Rishawi for the pilot. But Jordan's key condition is proof that al-Kassasbeh is still alive.
"At this point, we want to emphasize that we have asked for proof of life, and we have not received anything as of yet," government spokesman Mohammed al-Momani said Thursday afternoon, about an hour before the purported deadline.
The lack of proof of the pilot's health is concerning, but Al-Kasasbeh said the family had received some information from "unofficial sources" that was comforting.
Kidnappers are in a commodities exchange business, said Chris Voss, a former FBI hostage negotiator. And "proof of life is part of this type of negotiation."
"The Jordanians have no business going forward until they get it."
Al-Kassasbeh was captured after he ejected from his F-16 jet last month near Raqqa, the extremist group's de facto capital in Syria.
Coalition warplanes and drones continue flying over ISIS strongholds in northern Syria, looking for any signs of unusual activity -- perhaps a convoy moving toward the border with Turkey.
But hope is fading.
"Given the fact that ISIS has executed every hostage that they have publicly produced on a videotape except one British hostage, it does suggest that ISIS is not that interested in serious negotiations and is principally interested in the ability to get a lot of attention for its cause," said Peter Bergen, CNN's national security analyst.
The long wait
The 47-year-old Goto left Japan last fall, when his youngest daughter was 3 weeks old. His wife, Rinko, who heard from his captors first on December 2 and most recently in a Wednesday email with their "final demand" -- pleaded for his return.
"I hope our oldest daughter, who is just 2, will get to see her father again," his wife, Rinko, said.
Thursday's deadline was the third that ISIS set in less than a week.
The wait began with a video showing two Japanese hostages, Goto and aspiring security contractor Haruna Yukawa. And an outrageous demand from ISIS: a $200 million ransom from Japan within 72 hours, or else both captives die.
The first deadline passed a week ago Friday. Grim proof soon followed: An image of Goto holding what appeared to be a photo of Yukawa's headless body.
Goto apparently was spared.
"What it shows is that they are reacting tactically," said CNN military analyst Ret. Lt. Gen. Mark Hertling. "They no longer have a strategic plan. ... They are just trying to get the best thing going."
Days later, came a change in demands: ISIS now sought the release of al-Rishawi, a female jihadist imprisoned in Jordan for her role in a 2005 suicide bombing at a wedding reception that killed dozens. And if the swap wasn't made, the group threatened to kill both the pilot and the journalist.
Wednesday night brought yet another deadline: deliver al-Rishawi to the Turkish border by sunset Thursday, or al-Kassasbeh and Goto would die.
The relentless back-and-forth has raised hard questions about whether ISIS is truly negotiating, and whether Jordan made a mistake to try.
"They are attempting to elevate themselves into some sort of status as a political movement and a state," Hertling said. "They are not. They are terrorists."