The right wing doesn’t waste its time attacking hopeless Democratic presidential candidates. So this week’s spate of stories claiming that Elizabeth Warren had lied about being fired from a teaching job when she was pregnant in 1971 was a perverse compliment. Warren is now seen as a real threat to the reelection of President Donald Trump.

“What’s encouraging, and what has surprised me, is the blowback,” says Jennifer Palmieri, who was the communications director for Hillary Clinton’s 2016 campaign. “Not just from Warren’s campaign, but on social media, and even reporters defending Warren’s account of the pregnancy story. And a lot of the coverage has pointed out these attacks are being made in bad faith by outlets with a clear agenda. You wouldn’t have seen that four years ago.”

Warren—and Kamala Harris, responding to attempts to draw her into a controversy at her husband’s law firm—deftly used the attacks to highlight the fact that sexism is alive and well. The two episodes also underscore an enormously important question for all the Democratic contenders: Who can best turn out women voters in 2020, particularly those independents who helped swing tight contests to Trump four years ago?

Unfortunately for Warren and Harris and Amy Klobuchar, when it comes to attracting female voters there hasn’t been much advantage in being a female candidate. “There is very little evidence for a gender-affinity effect,” says Kathleen Dolan, a political science professor at the University of Wisconsin-Milwaukee who has studied the subject. “Partisan and racial affinity are much stronger. Though if you look at independent voters and gender, young women are the least likely voters to turn out. If that group is energized, it could have an impact.”

Cornell Belcher, a pollster for both of Barack Obama’s winning White House runs, knows the history too. “The Democrats haven’t won white women in a presidential election since 1996,” Belcher says. “The gender gap was driven by women of color.” But he is spotting signs of a change, largely as a reaction to Trump. “Obama didn’t win white college-educated voters, nor did Hillary. But in the 2018 midterms, Democrats all of a sudden began to eke out a majority with them. And 2018 was absolutely the year of the woman, as candidates and as voters. That is a potential party realignment.”

A FiveThirtyEight analysis of national polls in early September showed Warren with the greatest gender edge among Democratic candidates, but even that was slight, at 2.9 points. In Iowa, a mid-September survey of likely caucus attendees showed Warren ahead of Joe Biden by 2 points with women; in New Hampshire, among women, she lags Bernie Sanders by 2 points; in South Carolina, Biden clobbers Warren with women, by 27 points. The spread is even wider with a key demographic: South Carolina black women Democrats favor Biden over Warren by 47%–8%. “I don’t put a lot of weight on that as yet,” Belcher says. “The electorate is very fluid. They are supporting the candidate they have a deep background and roots with, who is Biden. The same was true with Hillary Clinton in 2008. Warren hasn’t spent campaign dollars in a real way yet to tell her story.”

When it comes to issues, female voters are hardly monolithic, but Warren has advocated a number of policy issues that could resonate: Among her many plans are protecting abortion rights, establishing universal childcare, and narrowing the pay gap for women of color. Plenty of women voters will decide the Massachusetts senator is too far to the left. But it is biography that gives Warren an opportunity—one that has been given a backhanded boost by the attacks on her early-life job history.