SULAYMANIYAH, Iraq - This most liberal of Iraqi cities, nestled in the mountains of Iraqi Kurdistan, is a perfect place from which to ponder the impact of the upcoming military defeat of the Islamic State - on Iraq and the world.

Suli (its nickname), a low-rise city with a population of around 2 million is home to the American University of Iraq, Sulaymaniyah (AUIS), a modern, private co-ed campus, where young men and women (most with hair uncovered) argue unimpeded with their professors in a manner light years away from the rote learning of so many Arab universities.

AUIS also hosts the annual Sulaimani Forum, a high-level gathering of Arab, Kurdish, and Western scholars and experts that is wrestling this year with a billion-dollar question: How can a fractured country and region prevent a new jihadi upsurge after the Islamic State is defeated?

That question is far from academic. Just four hours away from Suli, the battle to liberate Iraq’s second-largest city, Mosul, from the jihadis, is entering its final stage. It is being conducted by Iraqi troops and Kurdish Peshmerga fighters, many retrained by U.S. forces, who are also providing air power.

“Iraqi troops are moving toward the center of Mosul city, which is on the verge of being totally liberated,” said Jabar Yawar, spokesman for the Kurdish Ministry of Peshmerga, at the jam-packed gathering. “However,” he added ominously, “Daesh (an Arabic acronym for the Islamic State) won’t be ended militarily or internationally after Mosul is liberated.”

However weary Americans are of Iraq, they can’t afford to ignore Yawar’s warning. Not if they want to avoid the arrival of the Islamic State 2.0.

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Before I explain what that means, let me give you the good news.

One year ago, when I visited Iraq, skeptics doubted that the Iraqi military could crush the jihadis. That military, whose professional officer corps had been undermined by a corrupt government, had collapsed in the face of the Islamic State’s surprise 2014 invasion from its redoubts in Syria. But having regrouped (with U.S. help), those Iraqi and Kurdish forces have steadily pushed the Islamic State out of key parts of the country.

And now, spearheaded by the U.S.-trained Counter Terrorism Service (CTS) - the Iraqi military has pushed well into Mosul, the economic heart of the Islamic State’s so-called caliphate.

“ISIS considers the (territorial) caliphate at the center of their movement, and its fall means the collapse of the movement,” the burly CTS commander, Lt. Gen. Talib Shaghati, told the conferees, using another name for the Islamic State. Iraqi forces, and their U.S. air support, have carefully avoided using the kind of heavy fire that would cause civilian casualties. (Compare that with the deliberate bombing and shelling of civilians used by the Russian and Syrian militaries in Aleppo.) And the general spoke of strict punishment for revenge-seeking troops.

“We provide a live example of competence to civilians,” Shaghati said proudly. His force, which includes a broad ethnic and religious mix, is a stunning example of how patriotism can unite the complex fabric of Iraqi society. Indeed, at the military level, the fight to vanquish the Islamic State has acted like a glue - enabling cooperation among frequent antagonists.

What happens when the fighting stops?

Most Iraqis worry that the same grievances that allowed the Islamic State to flourish will enable jihadi cells to regroup and reemerge. The history is there: Sunni anger at being marginalized by the U.S. invasion and by the sectarian Shiite-led governments produced al-Qaida in Iraq in the mid-2000s, and its successor group, the Islamic State.

Today, Sunni cities lie in ruins, their infrastructure deliberately destroyed by the jihadis or damaged by fighting. The Sunni community is divided between backers and supporters of the Islamic State, and all fear vengeance by Shiite militias.

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Iraq’s moderate Shiite prime minister, Haidar Abadi, called for reconciliation at the Suli conference, but he heads a government financially strapped by the war and low oil prices, and undercut by Iranian meddling. Unless he can find funds to rebuild wrecked cities, the angry and unemployed may again find purpose in jihadist ideology. If the country splinters further, if Iranian mischief precludes reconciliation, watch out.

And yet, at the Suli conference, there were glimmers of hope.

Over and over, Iraqis talked of their weariness of war (although some pockets of the Islamic State remain to be liberated). While their civilian government has shown none of Shaghati’s competence, Iraqis would be relieved if Abadi could start the rebuilding. That might give him credibility to rally Iraqis above sectarianism and to curb sectarian militias backed by Iran. It might inspire new political parties for 2018 elections that aren’t based on ethnicity and religion.

Here is where the United States comes in. While many Iraqis have written off the USA as isolationist (early excitement at the Trump victory has faded), there is still a yearning for a U.S. presence, including military trainers. I was told repeatedly that this presence is vital to balance Iran and to help mediate sectarian disputes. Washington can also play a critical role in rallying international aid for rebuilding.

The cost of fighting the Islamic State has been too high to permit another jihadi movement to take root. Having broken Iraqi in 2003, the United States cannot afford to ignore it.

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One last note: Shaghati was denied a visa to visit family members in America. Now that the Trump team finally dropped the foolish visa ban against Iraqis, it should welcome the man who embodies the dream of a unified Iraq.

Trudy Rubin is a columnist and editorial-board member for the Philadelphia Inquirer.