Sen. Kirsten Gillibrand (D-N.Y.) in April. (AP Photo/Mary Altaffer)

In the up-down race for control of Congress, Democrats appear to have a noticeable advantage on one of the most high-profile issues of the moment: immigration overhaul. But they're at real risk of undercutting that when key members of the party embrace a left-wing cause: abolishing Immigrations and Customs Enforcement.

As his party struggles to find a consensus on a number of immigration issues, President Trump often steps out of the realm of reality to blame Democrats, particularly by claiming they won't help because they want “open borders.”

It's a factually incorrect statement, for sure. In 2013, a Democratic-controlled Senate passed an immigration bill that significantly beefed up border security, and today, some Senate Democrats are open to funding Trump's wall along the U.S.-Mexico border as part of a broader immigration deal.

But Democrats aren't helping make the distinction between outlandish political rhetoric and their actual positions when their left wing calls for abolishing the federal agency best known for deporting illegal immigrants. That's what new liberal darling Alexandria Ocasio-Cortez did in her remarkable upset of 10-term Rep. Joe Crowley (D-N.Y.) this week. A few days later, one of Democrats' most prominent and influential voices, Sen. Kirsten Gillibrand (D-N.Y.), joined in.

In an interview Thursday evening on CNN, the potential 2020 presidential contender said she agrees with escalating calls on the left for getting rid of ICE. “I agree with it. I don’t think ICE today is working as intended,” Gillibrand told CNN's Chris Cuomo.

She went on: “I believe that it has become a deportation force, and I think you should separate the criminal justice from the immigration issues, and I think you should reimagine ICE under a new agency with a very different mission and take those two missions out.”

Notice Gillibrand didn't say that the United States should stop deporting illegal immigrants altogether. She just said this particular agency should stop doing it because it has become too controversial.

That's not far from what 19 ICE agents argued recently in a letter to Homeland Security chief Kirstjen Nielsen that the Texas Observer uncovered. They claim that the drama over deportations is hindering all the other work ICE does, and they asked to have their agency spun off from the main one.

Proponents of abolising ICE don't necessarily want to open all borders to illegal immigrants and let them stay forever. But those are nuances that will likely be completely missed — or ignored — by the president and his allies as they try to cast Democrats as the reason Washington can't get anything done on immigration reform.

The irony here is that Democrats appear to have the upper hand right now on which party voters prefer to handle immigration. And Trump may have been helping them.

A Pew Research Center poll released just before the recent furor over family separations at the border found that Democrats had a record-high 14-point advantage over Republicans in voters' trust to handle immigration. That same Pew poll found that 48 percent of registered voters say they favor the Democratic candidate in their district, while 43 percent favor or lean toward the Republican.

Other polls showed two-thirds of Americans oppose Trump's family-separation policy, which the president underscored was his when he stopped it with an executive order. Under Trump, argued The Post's James Hohmann in a recent Daily 202 newsletter, Americans are actually becoming more supportive of immigration, not less.  That's despite the fact that an early June Fox News poll (done before the border drama) found Republicans held a 14-point advantage in trust to handle border security.

Amid all this, abolishing ICE has been a position taken by only the far left of the Democratic Party. Ocasio-Cortez is one of the cause's ringleaders, and she is a member of Democratic Socialists of America.

Now, a likely Democratic presidential contender just owned the idea of abolishing ICE. That could make it that much more difficult for Democrats to get out from under Trump's false claims that Democrats actually welcome illegal immigration.

Scott Clement contributed to this report.