It is now only a matter of weeks or even days before the villages are overrun and the Islamic State’s vaunted state-building enterprise in Syria and Iraq is brought to an end, military officials say.

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The conclusion of the 4½ -year war will also add urgency to the question of when and how the United States will pull its forces out of Syria, as ordered by President Trump last month. He later said they would remain to finish the fight against the Islamic State, and still no date has been set for their withdrawal.

Col. Sean Ryan, the U.S. military spokesman in Baghdad, cautioned that the end of the war on the ground will not halt the threat posed by the Islamic State, which is trying to regroup as an insurgent force in many of the areas where it has lost control.

Although the military has withdrawn some equipment, no troops have left and no deadline for a pullout has been issued, he said. In the meantime, U.S. troops will be focusing on training their local partners and stabilizing the area to deter a return of the militants.

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“The end is near, territorial-wise, but they will continue to regroup and, like we’ve seen over the past two weeks, will try to create spectacular events,” Ryan said, referring to the suicide bombing at a restaurant in the town of Manbij this month, which killed four Americans. It came more than three years after the Islamic State had been driven out of that area.

The holdouts in the two desert villages include some of the most committed extremists, who have remained on the battlefield despite multiple opportunities to escape or surrender and can be expected to fight until the end.

But if the campaign by U.S.-backed forces proceeds as well as it has in recent days, it could be over within as little as two weeks, Ryan said. Mazloum Kobani, the top commander of the SDF, told Agence France-Presse on Friday that it would take no more than a month.

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“I believe that during the next month we will officially announce the end of the military presence on the ground of the so-called caliphate,” Kobani said.

The remaining militants include some of the Islamic State’s most senior leaders and “famous terrorists,” according to Zana Amedi, a spokesman for the Kurdish People’s Protection Units, or YPG, which is the biggest component of the SDF.

U.S. and Kurdish officials do not believe, however, that Abu Bakr al-Baghdadi, the Islamic State leader, is among them. “He would have to be stupid to stay in the last pocket until the end,” Amedi said. “He would have escaped a long time ago before the situation deteriorated for them so much. He’s not dead, and he’s not in these villages, either.”

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A big percentage of those who remain are foreigners who flocked to Syria to join the Islamic State when it was at the peak of its power. Because of their appearance, they would stand out among the locals if they tried to blend in with the civilians fleeing the battlefield, as some local fighters have done.

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“They have no choice but to fight to the death or surrender,” Amedi said.

Dozens of foreigners are among scores of fighters who have surrendered or have been caught trying to escape in recent days, Amedi said. A majority of the foreigners are Iraqis, but there are also Europeans, Americans and Asians among them.

Thousands of civilians have also been streaming out of the area and handing themselves over at SDF checkpoints, where they are screened for Islamic State fighters trying to hide among them, he said.

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Civilians have been dying too — in the crossfire, at the hands of Islamic State snipers who seek to prevent them from escaping and in the U.S.-led airstrikes providing the muscle behind the offensive.

In one of the worst incidents, as many as 39 civilians were alleged to have been killed in Baghuz earlier this month when vehicles in which they were trying to flee were struck by coalition warplanes. In a separate incident, the United States said Friday that it had launched an investigation into the death of a mother and the injury of two of her children when their car was allegedly hit by an airstrike.

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The eventual fate of the entire swath of eastern Syria now under U.S. military control as a result of the Islamic State war has yet to be decided. Trump’s sudden announcement that the troops would be withdrawn set off an international scramble to figure out what to do with the territory, which amounts to almost a third of Syria.

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U.S. officials visited Ankara this month to discuss Turkey’s demand for some form of “safe zone” in northeastern Syria. The Kurds in the U.S.-allied YPG are aligned with the Turkish PKK guerrilla group, and Turkey has said it will deploy troops across the border into Syria to prevent Kurds from launching cross-border attacks once the U.S. troops have gone.

The Kurds, however, fear persecution by the Turks should Turkey assume responsibility and have turned to Russia for help in brokering a deal that would give the Syrian government overall control of the area.

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Turkey is also talking to Russia, which has proposed restoring Syrian government control under the terms of a 1998 agreement between Turkey and Syria. That accord required Damascus to guarantee that Kurds would not launch attacks against Turkey from Syria.

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Any vacuum of authority in the area would offer an opportunity for the Islamic State to reestablish itself among the local populace, whose loyalties are deeply divided among the multiple powers competing for control, said Nicholas Heras of the Center for a New American Security.