For weeks, Iraqi forces backed by U.S. and coalition air power have moved ever closer to the city of Mosul, the heart of the Islamic State terror group's holdings in Iraq.
Yet for all the progress, these forces would appear to be no closer to cutting off the group's head, the elusive cleric Abu Bakr al-Baghdadi.
"We don't know where he is," Operation Inherent Resolve spokesman Col. John Dorrian told Pentagon reporters via a videoconference from Baghdad last week. "If we knew where he was, he would be killed at once."
In the place of certainty, there is rumor.
In recent weeks, Iraqi military officials have insisted that the 45-year-old Baghdadi was hiding in tunnels and bunkers in Mosul, directing the defense of the city himself.
More recently, reports from Iraq quote various sources as saying Baghdadi is in Mosul preparing for his own death and in the process of selecting a replacement.
Western intelligence officials are highly skeptical. They say that even though IS and its predecessor groups have strongly embraced martyrdom — to the extent that some see it as a sort of death cult — the group's leaders have consistently opted to live to fight another day.
Where is he?
Just last week, British Foreign Secretary Boris Johnson said the most recent intelligence on Baghdadi suggests he has "vacated the scene" in Mosul.
And pinpointing his current whereabouts will be a tall order, with some suggesting the IS leader is more skilled at evading detection than even al-Qaida founder Osama bin Laden.
"He has an intense OpSec [Operational Security] posture," said Michael Smith, co-founder of Kronos Advisory, a private intelligence firm, who has served as a contributing expert to the Congressional Task Force on Terrorism and Unconventional Warfare.
Smith and others say IS has leveraged its own experiences with high-level espionage techniques from Iraqi intelligence officers once loyal to former dictator Saddam Hussein, making the group a potent and sophisticated adversary.
"They keenly understand our tech capabilities," Smith said. "They have a very strong sense of our targeting tactics and how to flout them."
Ready to shoot
U.S. officials say, if and when they pick up Baghdadi's trail, they will not hesitate to take a shot.
"Anytime you can take someone like that out, it always has impact," one official told VOA.
Yet there are some who argue that not killing Baghdadi, at least not yet, may actually help in the efforts to destroy IS.
"The second- and third-order effects of prematurely targeting Baghdadi are too great," according to Nicholas Glavin, a senior researcher at the U.S. Naval War College's Center on Irregular Warfare and Armed Groups. "He would be seen to his supporters as the caliph who was martyred for the sake of the caliphate."
But if Baghdadi is alive to see the remnants of his self-declared caliphate fall to U.S.-backed forces, the impact could be lasting.
"This would be a precision strike at the group's ideology, a feat that has been quite difficult to date for the coalition," Glavin said.