“I will do everything I can to help him win and save this country,” Governor Noem of South Dakota said Friday evening of President Trump, endorsing his re-election bid at a Rapid City rally where the candidate hailed her as “one of the most successful governors in the entire nation” and said her backing “means a lot.”

Ms. Noem’s endorsement follows comments she made on Newsmax Thursday, when, asked whether she would consider signing on to Mr. Trump’s ticket, she replied, “Absolutely,” adding, “I would in a heartbeat.” Earlier this week, Rolling Stone reported that a former TV news anchor, Kari Lake, and Congresswoman Marjorie Taylor-Greene of Georgia are locked in a “death race” to become Mr. Trump’s vice presidential pick.

Developments like these are fueling speculation that Mr. Trump, the presumptive GOP presidential nominee, could choose one of these women as running mate in the 2024 election. The thinking goes that it could help mitigate Republicans’ perennial underperformance with women and maybe work to stem the electoral bleeding that the Supreme Court ruling overturning Roe vs. Wade appears to have started.

While no one knows what Mr. Trump is thinking, it’s clear that these three women, who often top the list of female vice presidential contenders, are interested in the job. Yet it’s not certain that the vice presidential pick will have that much of an effect on the election.

Aside from Ms. Noem’s confirmation of interest on Thursday, Ms. Greene, the Georgia congresswoman, has said she would “very, very heavily” consider the offer if it were extended, and Ms. Lake took up de facto temporary residence at Mar-a-Lago for months to try to win favor from Mr. Trump, according to Vanity Fair.

It’s unclear how much of a difference having a woman on the ticket would even make. On the presidential level, it’s far from clear that a woman running mate has the potential to turn an otherwise losing campaign into a winner.

According to historical exit poll data compiled by the Rutgers Eagleton Institute of Politics, the gender gap — referring to the difference between how men and women vote — emerged in the 1980 election when men broke heavily for President Reagan while women voted nearly evenly for President Carter and Reagan.

Since then, the gap has only grown, with women as a group backing Democrats in every election since 1988. A political scientist at John Jay College, Brian Arbour, tells the Sun that this isn’t because Democrats put up more female candidates but because of the evolving politics of the Republican Party.

“In general women are more liberal, and so the conservative turn in the Republican Party had more purchase among men than among women,” Mr. Arbour says.

Mr. Arbour says he is skeptical that choosing a woman as the vice presidential pick will have much of any effect on the outcome of the election. That’s because he says there’s not a lot of evidence that voters spend much energy considering the vice president. Rather, they tend to focus their attention on the top of the ticket.

“The only exception to that is that in 2008 Sarah Palin managed to get distinctly negative opinions from Democrats,” Mr. Arbour says. “If Trump is the nominee, the vice presidential nominee will not be the center of discourse.”

With that being said, there is some anecdotal evidence that, on the state level, tickets with some gender diversity do seem to perform fairly well.

Ten states choose their lieutenant governors in what’s called “presidential style,” meaning that the gubernatorial nominee chooses their number two after the primary elections, Florida, South Carolina, New Jersey, Indiana, Michigan, Iowa, Kansas, South Dakota, Nebraska, Colorado

Of these states, only Nebraska has both a governor and lieutenant governor of the same gender, Governor Pillen and Lieutenant Governor Kelly.

While on the surface this might make it seem like having a gender-diverse ticket helped these officials win election, it could also be a case in which correlation does not equal causation. It could be a coincidence or it could mean that crafting a ticket with some diversity is the mark of a successful campaign.

This is an explanation to which Mr. Arbour is sympathetic, saying that “tickets are a way to unite coalitions in the party,” and that one of the ways to do so is picking a no. 2 who appeals to a different part of the party than does the person on the top of the ticket.

In 2016, Mr. Trump’s choosing Mike Pence was a good example of this because, at the time, evangelical Christians seemed to have only lukewarm support for Mr. Trump, and Mr. Pence served as an olive branch to that core constituency of the GOP.

The same dynamic is at play in 2024, which is one of the reasons that most factions of the party like Senator Scott, a Black man and devout Christian who has been mentioned as a possible vice presidential nominee.

This is a point emphasized by an associate editor at Sabato’s Crystal Ball at the University of Virginia Center for Politics, Miles Coleman, who tells the Sun that “having a balanced ticket is always good.”

Mr. Coleman says that choosing a woman might be a way to balance the ticket, even if the effect is a “little less tangible” than it is sometimes made out to be. “A lot of suburban women are going to vote against Trump no matter what,” he said. “Having a woman on the ticket might be something of an olive branch.”

Mr. Coleman says he isn’t convinced that any of the women drawing headlines, i.e., Ms. Greene, Ms. Lake, or Ms. Noem, make the most sense from a ticket-balancing perspective, because they are all already aligned and associated with Mr. Trump.

“Trump likes to be the center of attention,” Mr. Coleman said. “I wonder if someone like MTG or Kari Lake might distract from him.”

From a ticket-balancing perspective, Mr. Coleman suggested that someone who has been less of a lightning rod for scrutiny might be a better pick, such as Mr. Scott or, if Mr. Trump was to pick a woman, Governor Reynolds or Congresswoman Elise Stefanik.

Mr. Coleman says that Mr. Trump’s strategy should be similar to that of 2016: choosing a candidate who can shore up the flanks of his coalition. He adds the caveat that “with Trump, there’s the stuff he should do and the stuff he actually does.”

In terms of the horse race, Mr. Arbour says that the main thing the vice presidential speculation tells you is that “everyone thinks Trump is going to be the nominee.”