Originally Published: 22 JUN 22 13:06 ET

By Daniel Dale, CNN

(CNN) -- Mehmet Oz, the Republican nominee in Pennsylvania's US Senate race, has been claiming that everyone he defeated in the party primary has now endorsed him. But that's not true.

Oz is trying to unite the Republican base after winning the bruising multi-candidate primary with about 31.2% of the vote. In an appearance on Fox on Tuesday, after a host mentioned an opinion poll that showed Oz trailing his Democratic opponent, Pennsylvania Lt. Gov. John Fetterman, by 9 percentage points, Oz spoke of the importance of Republican unity and said, "All of the Senate candidates that I was competing against have endorsed me. The party rapidly unifies."

Oz made the same claim in a Fox appearance last week, saying then that "my competition -- they've all endorsed me, by the way. God bless them, gracefully done that."

Facts First: It is not true that "all" of Oz's former competitors have endorsed him. While Oz has been endorsed by three of the four other major candidates from the May primary -- second-place finisher David McCormick (who received about 31.1% of the vote), fourth-place Carla Sands (about 5.5%) and fifth-place Jeff Bartos (about 5%) -- Oz has not been endorsed by third-place Kathy Barnette, who received about 24.7% of the vote.

"No, I have not endorsed Oz. He knows that," Barnette said in an interview on Tuesday. She said she wasn't surprised "the truth may elude them sometimes" given how, she said, "they lied about me" during the campaign.

That's not all she had to say about Oz. She added that about 70% of Republicans "voted against him" despite his endorsement from former President Donald Trump. She argued that "Donald Trump won this election for Oz; it wasn't Oz." And though she said that she was open to eventually endorsing and campaigning for Oz, she also said, "I have a lot of issues with Oz and who he presented himself to be."

When CNN noted to Oz's campaign earlier on Tuesday that Barnette has not endorsed Oz, the campaign didn't try to argue that Oz's "all" claim was correct. Instead, spokesperson Brittany Yanick said simply: "The Republican Party is uniting behind Dr. Oz."

Barnette's video

Barnette is a right-wing commentator who surged in the polls late in the primary campaign as Oz and McCormick focused on each other. Her rise prompted a flurry of deceptive attack ads from super PACs backing Oz and McCormick.

On June 6, three days after McCormick conceded the race and announced he was backing Oz against Fetterman, Barnette released a video on social media in which she made clear she was not endorsing Oz immediately.

Barnette said in the video that a lot of people are wondering "whether or not I'm going to endorse Mehmet Oz." Over the next four-and-a-half minutes, she explained that she believes Oz needs to work for her endorsement.

"I owe no one anything ... and I have no interest in just giving anything to anyone," Barnette said in the video. "No one certainly gave anything to me. I had to go out there and earn it."

Barnette said in the video that her fidelity is to her own 330,000-plus primary voters "and to them alone." She said it is Oz's responsibility to make sure that their voices, along with the voices of people who voted for other candidates, "are being heard, and that their voices are being reflected in his policies."

"Oz is gonna have to do the hard work of campaigning for himself," Barnette said in the video. "It's gonna take a village -- all of us, myself included, are gonna have to come out and support him -- but I believe it is his responsibility to make sure that he earns that support."

So: Barnette clearly left the door open to a possible future endorsement. (She said Tuesday, "I want my party to win" and that she is "very open to backing him and stumping for him.") But she was just as clear that she wasn't endorsing yet.

Barnette made bigoted anti-Muslim remarks in the 2010s; Oz would be the country's first Muslim senator. Barnette said Tuesday that she made these comments years ago, at what she claimed was "a very different time in our culture," and that although she has a lot of issues with Oz, "his religion is not one of them."