Democrats taking to the airwaves to denounce Judge Brett Kavanaugh have been considerably less focused on their own abuse investigation into Democratic Rep. Keith Ellison.

The Minnesota Democratic-Farmer-Labor Party said a month ago it would investigate after a former girlfriend accused Mr. Ellison of emotional and physical abuse, but since then the party has said nothing as the probe fades largely from view.

That’s undoubtedly by design, said Larry Jacobs, professor at the Humphrey School of Public Affairs at the University of Minnesota, who characterized the investigation as an exercise in damage control aimed at providing the party with political cover if more allegations erupt.

“I think of this not as an investigation to uncover facts and make a public release that would hurt the Democratic Party before the election,” Mr. Jacobs said. “That’s not what this is. This is closer to political insurance in case there are revelations that become a kind of October surprise for Ellison and then hurt the entire DFL ticket.”

Karen Monahan, the Sierra Club organizer who levied the abuse allegations Aug. 12, weighed in Friday by accusing state Democrats, including DFL chairman Ken Martin, of working behind the scenes to squelch support for her.

“It is not democratic for @kenmartin73 & others from the @MinnesotaDFL to call people and tell them to stop showing support for justice and truth in this case or threaten to take away contracts and resources if they show support,” she said on Twitter.

Ms. Monahan also said she planned to release more information, saying, “We are organizing and many stories will come.”

It is not democratic for @kenmartin73 & others from the @MinnesotaDFL to call people and tell them to stop showing support for justice and truth in this case or threaten to take away contracts and resources if they show support. We are organizing and many stories will come — Karen Monahan (@KarenMonahan01) September 21, 2018

The DFL did not immediately return a request for comment, but unnamed party officials told KSTP-TV in Minneapolis last week that the month-old investigation “remains alive.”

“We take these allegations seriously, we are certainly eager to hear all the answers to the questions that have been raised,” Mr. Martin said at an Aug. 16 press conference.

Mr. Ellison, who gave up his House seat to run for attorney general, has maintained a relatively low profile since the allegations went public, confirming that he and Ms. Monahan were involved in a relationship but denying the domestic abuse charges.

The campaign said in a statement that Mr. Ellison, who serves as Democratic National Committee deputy chair, is “cooperating fully with that independent investigation,” but Democrats did not explain what was meant by “independent,” according to KSTP.

Mr. Jacobs challenged the claim, saying, “This is not being done by an independent party.”

“This is being done by the party that benefits from the outcome of the investigation,” he said. “It’s conflicted from the very start by definition.”

Ms. Monahan said in Monday tweets that she had turned over medical and therapy records and agreed to be interviewed with Mr. Ellison in the room as part of the investigation, but that she had been “smeared, threatened, isolated from my own party.”

No, they don’t. I’ve been smeared, threatened, isolated from my own party. I provided medical records from 2017, stating on two different Dr. Visits, I told them about the abuse and who did it. My therapist released records stating I have been dealing and healing from the abuse — Karen Monahan (@KarenMonahan01) September 17, 2018

I told the investigator I would be happy to talk with @keithellison w/ them in the room. They asked Keith if he would be willing & he declined any meeting w/ me. — Karen Monahan (@KarenMonahan01) September 18, 2018

Kevin Poindexter, Minnesota Republican Party executive director, called the handling of the matter “hypocrisy at its finest,” given the Democratic calls for an FBI investigation into a 36-year-old allegation against Mr. Kavanaugh, the Supreme Court nominee.

“They’re always calling for independent investigations, but Ken Martin kept this in-house,” Mr. Poindexter said. “They’re running the investigation, they’re not releasing anything, and they’re slow-rolling this thing so they can get past the election so it doesn’t hurt Ellison and the rest of the candidates.”

Mr. Kavanaugh has denied a California woman’s allegation that he pinned her on a bed and tried to undress her when they were in high school, but Democrats such as New York Sen. Kirsten Gillibrand have said they believe the accuser.

The DFL has endorsed Mr. Ellison, whose presence on the ticket is expected to benefit candidates like Democratic Sen. Amy Klobuchar by drawing progressive voters — as long as the domestic abuse charges don’t come back to haunt the party.

The worst-case scenario for Democrats would be the release of a video that allegedly shows Mr. Ellison dragging Ms. Monahan from a bed and screaming at her. Ms. Monahan’s son Austin said last month on Facebook that he had seen the video, but Mr. Ellison has said no such incident occurred.

“Karen and I were in a long-term relationship which ended in 2016, and I still care deeply for her well-being,” Mr. Ellison said in a previous statement. “This video does not exist because I never behaved in this way, and any characterization otherwise is false.”

If such a video were to appear before the election, “it would be a disaster,” Mr. Jacobs.

“It would create a firestorm around Ellison and the entire ticket, forcing people like Amy Klobuchar to make statements and distance herself from Ellison, which would destabilize her race because Ellison has a lot of progressive supporters,” said Mr. Jacobs.

Ms. Monahan has said she will not release the video, calling it “humiliating,” but posted a November 2017 medical document earlier this week in which she told a clinic she suffered “emotional and physical abuse” during her relationship with Mr. Ellison.

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