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Faced with mounting problems for GOP presidential nominee Donald Trump, Utah Republican Party Chairman James Evans turned Thursday to a conspiracy theory popular among conservative media sites — that Bill Clinton fathered a child with a black prostitute in the 1980s.

Evans accuses the liberal mainstream media of covering up the dalliance, or at least ignoring it, and says that if news outlets are going to publish accounts about women who say they were groped or assaulted by Trump, they should pursue the story about Clinton.

Evans raised the question of the illegitimate child on CNN, drawing an incredulous response from anchor Carol Costello, who quickly shut down the interview.

Evans had been on the newscast to discuss Trump's collapse in Utah polls since video surfaced of Trump making lewd comments about women and bragging about sexual assault. Since then, several women have said Trump engaged in sexually aggressive and unwelcome behavior with them.

The list of Trump's accusers includes a former Miss Utah, Temple Taggart McDowell, who told NBC News on Wednesday night that, when she was 21 and representing the state in the Miss USA pageant, which Trump owned, he twice kissed her aggressively on the mouth.

"I remember being shocked. … I would have just thought to shake somebody's hand, and that was his first response with me," McDowell told NBC. "I remember feeling kind of embarrassed, like wanting to turn and wipe my mouth, like, 'What just happened?' "

But because he owned the pageant, she said, "I just kind of shooed it off and kind of went along."

Trump's poll numbers have fallen in Utah. The latest survey shows him with a narrow lead over Democrat Hillary Clinton and the surging independent candidacy of Provo native Evan McMullin.

Numerous Utah leaders, including Gov. Gary Herbert, Rep. Mia Love, Sen. Mike Lee and others, have abandoned Trump and even called for him to get out of the race altogether.

Evans continues to insist Trump will win Utah and, in an interview with The Salt Lake Tribune, doubled down on his assertion that the liberal media were ignoring the story about Bill Clinton's illegitimate child, just before holding a news conference with Utah reporters about the allegations of now-30-year-old Danney Williams.

"I remember about this child since he was 8 years old. It's not a hidden secret in the South," said Evans, who grew up in South Carolina and is black. "The point I was making was: All the media knows about it. [Costello] knows about it. … He's not showed up here three weeks before the election. He's been pursuing this for years, yet there's this wall of silence."

Williams was first pushed into the national spotlight when Bill Clinton was running for president. The story was that Clinton had paid Williams' mother for sex on a number of occasions, and in one of those encounters impregnated his mother. Williams' mother said that Clinton was her only white client.

There has been no proof offered to corroborate the story and Williams' mother, Bobbi Ann Williams, who did time in prison, lost custody of her son. Danney Williams has also done time behind bars.

Even so, the story got traction in conservative media outlets, particularly Drudge Report, and British tabloids. In 1999, a paternity test was reportedly conducted that — according to Drudge and others — showed Williams was not Clinton's son.

The Clinton DNA sample was based on tests conducted during the investigation into Clinton's relationship with Monica Lewinsky.

But the then-editor at the Star tabloid, which supposedly conducted the test, said the results were inconclusive.

Now, as Hillary Clinton's historic run for the White House is nearing the homestretch, Williams is resurfacing on social media and in conservative blogs.

"The fact that the media is giving voice to every other allegation out there against Donald Trump, they should give voice to all allegations, but there seems to be this protective wall around Hillary," Evans said. "All of these sorts of things need to be treated the same way, but there seems to be this one-sidedness."

And if tests conclude Clinton is the man's father, is it germane to his wife's fitness for the presidency?

"It would just give a more complete picture of the Clintons," Evans said.

Utah Democratic Party Chairman Peter Corroon said it's absurd to think the Clintons haven't been thoroughly vetted, calling them the most scrutinized couple in political history. He said Evans' broadside was "out of bounds."

"He's been spending too much time in the basement on the internet looking for conspiracy theories," Corroon said. "When the ship is sinking, you grab at anything to keep you afloat, and right now Donald Trump's ship is sinking, and, for the first time in over 50 years, we may have a Democrat winning the presidential [election] in Utah, and I don't think he can fathom that happening."

The progressive-leaning Alliance for a Better Utah called on Evans to resign Thursday, saying "we expect more of our politicians, especially leaders of our major political parties."

"Evans' constant attempt to spread rumors and lies, including ones that have been thoroughly proven wrong, are inappropriate for his position," said Rachel Sanders, the alliance's executive director. "Because of our tremendous respect for the multiparty system, we call on leaders and members of the Utah GOP to demand James Evans resign from office, so decorum and respect can return to Utah politics and the rest of this election season." Twitter: @RobertGehrke