An odd dynamic is developing in this first phase of the 2020 presidential race. Even as several female Democratic candidates, including Sens. Elizabeth Warren and Kamala Harris, gain traction, Democratic voters and party officials alike are fretting over the wisdom of putting another woman at the top of the ticket.

Needless to say, these doubts grow out of Hillary Clinton's still-shocking 2016 defeat. But to what extent was Clinton's failure due to sexism? Will a future female candidate with less political baggage begin the race with a significant disadvantage?

A recently published study offers mixed news for the women eyeing the Oval Office in 2020. It concludes that voters "do not seem to hold female candidates in disregard. All else equal, they prefer female to male candidates."

Still, voters also "reserve their highest reward for women who can both do the job of a politician and that of a wife and mother," writes a research team led by political scientist Dawn Langan Teele of the University of Pennsylvania. "Female candidates have to be superwomen, while male candidates enjoy the luxury of delegating family work to others."

When President Trump noted that he had never changed a diaper, people shrugged. Imagine the reaction if Sen. Amy Klobuchar made that claim.

There is no question that female candidates have to be very careful in how they present themselves so as not to trigger harmful stereotypes. Research has consistently found that voters still equate leadership with masculinity, and, likely as a result, they approach female candidates warily, giving them less latitude than their male counterparts.

For instance, one study reported that voters are less willing to forgive women for straying from party orthodoxy. Others have found that female, but not male, candidates get punished for being seen as power-hungry, or for having a gender-ambiguous look.

The study, published in the American Political Science Review, analyzed the results of three original surveys: One of 2,144 voters, and two of state legislators (nearly 3,000 respondents in all). Respondents were asked to choose between pairs of hypothetical political candidates. The potential office-holders' attributes — gender, age, number of children, occupation, number of years in politics, and spouse's occupation — were randomly assigned.

Based on the participants' choices, "We find no evidence of outright discrimination, or of double standards," the researchers write. "All else equal, most groups of respondents prefer female candidates, and evaluate men and women with identical profiles similarly. But on closer inspection, all is not equal," they add. "Voters and political elites prefer candidates that are married with children."

This preference, they argue, is problematic in at least two ways. First, "within the pool of professionals in political 'feeder' careers like business and the law, women are much less likely to be married than men," they write. These unmarried candidates may "struggle to connect with voters that implicitly favor a different candidate profile." Second, women who do have families are widely expected to devote considerable energy to maintaining the household and raising children. This can put female candidates at a serious disadvantage.

Those "who strive to be perfect wives and mothers [find themselves] without hours in the day to go head-to-head with their male counterparts," the researchers write. But those who cut back on traditional household duties to concentrate on their campaigns — or on the work of governing — "bear the burden to convince the public they can do both jobs well."

Of course, these gender stereotypes can shift over time; the researchers note that a childless woman, Angela Merkel, is Germany's most successful politician of the last two decades.

"But until then, the odds are against female candidates," the researchers conclude. "If women have the characteristics that voters and other public officials prize, they likely have too little time to actually take on the job."

This story originally appeared as Voters expect female politicians to be perfect mothers too on Pacific Standard, an editorial partner site. Subscribe to the magazine's newsletter and follow Pacific Standard on Twitter to support journalism in the public interest.