Even if President Trump stops calling her “Pocahontas,” Sen. Elizabeth Warren may be unable to silence the chatter over her dubious claims of Cherokee ancestry for at least the next year, thanks in part to someone whose Indian heritage isn’t in doubt.

Shiva Ayyadurai, a Bombay-born, MIT-educated entrepreneur running as an independent to unseat the Massachusetts Democrat, already is the early favorite for the November election’s catchiest campaign slogan: “Only a real Indian can defeat a fake Indian.”

In June, he sent Ms. Warren a DNA testing kit for her birthday. She sent it back, but the exchange exploded on social media.

“That’s the nightmare for Elizabeth Warren, because I can attack her from the left,” Mr. Ayyudurai said. “I can say, ‘Wait a minute, what do you know about the journey of a person of color? You know nothing. This is all theory to you. Because you did lie. You took advantage of affirmative action.’ “

It’s been nearly six years since Ms. Warren won the Senate seat in spite of her “fake Indian” problem, but the issue is undergoing a fresh round of scrutiny as the progressive standard-bearer looks ahead to her re-election bid and potentially the 2020 Democratic presidential primary.

Mr. Trump re-energized the issue last week by referring to her as “Pocahontas” during an Oval Office ceremony honoring Navajo Code Talkers. She countered by accusing him of using a “racial slur.”

“He thinks that somehow he’s going to shut me up with that, and it’s just not going to happen,” Ms. Warren told CNN. “It didn’t work in the past; it’s not going to work in the future.”

The issue wasn’t strong enough to defeat Ms. Warren in 2012 — she beat Republican Sen. Scott Brown by 54 percent to 46 percent — but one difference is that she is now being challenged on both the left and right.

Comedy Central’s “The Daily Show” host Trevor Noah weighed in last week with a monologue mocking her Cherokee claims as well as the crab recipe she submitted to the 1984 Cherokee cookbook “Pow Wow Chow,” which she has been accused of copying from the New York Times’ wire service.

“That would be like finding out I’m completely white, I have no African blood, and yet I wrote the book ‘Snacks for Blacks,’” the comedian deadpanned.

PolitiFact re-examined the issue in a Friday analysis, concluding that her “genealogical claim has zero documentary evidence to back it up,” although “there is no proof Warren gained any special advantage in her career.”

“Elizabeth Warren might never live down the charge of falsely claiming Native American roots,” PolitiFact said. “The controversy emerged during the Democrat’s successful Senate run in Massachusetts in 2012, and it continues to dog her.”

Her critics say she has only herself to blame for failing to put the matter to rest by apologizing for misrepresenting herself or at least acknowledging her mistake.

“Elizabeth Warren needs to apologize and admit that she knowingly claimed an employment status she was not entitled to claim,” said Cornell Law School professor William A. Jacobson. “Ignoring her own conduct, and pretending that those who criticize her are the problem, has worked so far for her in Massachusetts, but I don’t think it will work if Warren has presidential ambitions.”

Lest anyone forget the details, Mr. Jacobson created a website in 2013, “Elizabeth Warren Wiki,” that documents how she began identifying herself as American Indian as she climbed the faculty ladder at the University of Pennsylvania and then Harvard law schools.

In 2012, the New England Historic Genealogical Society found no proof that Ms. Warren was Cherokee, but she pointed to her family’s oral history, saying that “being Native American has been part of my story I guess since the day I was born.”

Referring to a photo of her grandfather on the mantle, she told local television reporters that her aunt Bea “remarked that her father, my pawpaw, had high cheekbones, ‘like all of the Indians do,’ because that’s how she saw it.”

Death certificates on file with the state of Oklahoma indicated that her mother and her aunt were white, according to Cherokee Nation genealogist Twila Barnes.

Harvard officials have said Ms. Warren’s heritage wasn’t a factor in the decision to hire her, but Peter Wood, president of the conservative National Association of Scholars, said it “seems very likely that Elizabeth Warren’s claims to Cherokee descent made her candidacy more attractive.”

“Harvard, like almost all colleges and universities, hides the degree to which racial preferences play a role in hiring and admissions,” Mr. Wood said. “I would be surprised if Harvard ever came forward to acknowledge that Sen. Warren’s purported ethnicity played a part in her appointment. So we are left with a conjecture, but a very plausible one.”

Claiming Cherokee ancestry, and later being lauded as Harvard Law School’s “first woman of color,” conflicts with her message that “the system is rigged against the average person,” Mr. Jacobson said.

“Yet Warren tried to rig the system in her favor by claiming a minority status to which she was not entitled,” Mr. Jacobson said. “I don’t think voters nationally will forgive that without a heartfelt and unconditional apology.”

Republican strategist Mike McKenna predicted Ms. Warren will get a pass if she runs in the 2020 Democratic presidential primary race, which he said has all signs of becoming “a contest of who will get farthest, fastest, and most emphatically to the left.

For Mr. Trump to call Ms. Warren “Pocahontas” is “a gift to her,” he said. “It reminds the crew that she was a target early on. And no one is ever going to check the story. So she won’t need to apologize.”

Ms. Warren felt confident enough about her footing to refer to the “Pocahontas” jab in a Monday fundraising plea.

“You might have heard that Donald Trump likes to call me ‘Pocahontas.’ He does it on Twitter, at rallies, and even in official White House meetings,” she said in an email as reported by the Boston Herald. “Let’s show Donald Trump that we’re sick of his racist slurs by getting to work to fight his agenda.”

Still, Mr. Noah told audiences Tuesday that Mr. Trump was “hitting Elizabeth Warren for saying she was Native American when she wasn’t — something she’s never apologized for or owned up to.”

“Elizabeth Warren did something problematic, the kind of thing that we call rightfully each other out for every single day,” Mr. Noah said. “So as weird as it is to say, in his own racially offensive way, Donald Trump was being ‘woke.’ And that’s unfortunately the truth.”

Mr. Ayyadurai may be a long shot, but he is not going away anytime soon. By running as an independent, he said, he has to gather only 10,000 valid signatures to qualify for the general election ballot, meaning he could be a thorn in Ms. Warren’s side until Nov. 6.

For his birthday, Mr. Ayyadurai is hosting a “DNA Dance Party” on Dec. 12 in Cambridge. Ms. Warren, of course, is invited.

“She’s got a major problem with me, I’m telling you, because I can expose her for what she is,” Mr. Ayyadurai said. “What Trump did is not racist. She’s the racist. … She shoplifted some other group’s race identity, and then when she’s exposed on it, she calls other people racist. And it is a legitimate issue.”

• Douglas Ernst contributed to this report.

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