Donald Trump once boldly claimed that he could shoot someone on Fifth Avenue and not lose voters. In the decade since, as Mr. Trump has persisted through scandals, controversies and an array of thorny challenges, I have been asked the same question over and over again in my capacity as a pollster: Will this be the thing that costs him his supporters?

It almost never is. Save for a brief moment after Jan. 6, 2021, Mr. Trump’s support from his base has been rock solid.

But the recent turmoil over the files on the disgraced financier Jeffrey Epstein — including the truly horrific sex crimes Mr. Epstein was convicted and accused of, the strange secrecy around it all, the conspiracy theories that flourish — people ask, might this be different? Is this the one?

So far, there’s little evidence to show that the fixation on the president’s past relationship with Mr. Epstein is taking a significant political toll. Polling by The Wall Street Journal conducted shortly after the Epstein story blew up again recorded Mr. Trump’s job approval at the same level, 46 percent, that it was in its polling in April. Concerns like cost of living, immigration and the state of the economy remain dominant among Americans.

At the same time, while the Epstein saga has done little to dent Mr. Trump’s overall popularity, it is perhaps the starkest example of a schism between the president and his most loyal supporters.

A growing volume of data shows that Mr. Trump’s own voters nevertheless harbor concerns about how the Epstein issue is being handled. While Republicans routinely give Mr. Trump favorable job approval ratings on a wide range of issues, a recent Quinnipiac poll showed that only four in 10 Republicans specifically say they approve of his administration’s handling of the Epstein files. In a Washington Post poll published this week, only 38 percent of all Republicans — and 43 percent of MAGA Republicans — approve of the president’s handling of the Epstein files.