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The first poll of likely New Hampshire Democrats since last week’s debates shows that penetrating the top tier of three candidates might be tougher than ever in the state’s first-in-the-nation primary.

With six months until the anticipated February 2020 contest, a Suffolk University/Boston Globe poll released Tuesday found former vice president Joe Biden the leader among likely Democratic primary voters with 21 percent. In second, Senator Bernie Sanders of Vermont had 17 percent, and, in third, Senator Elizabeth Warren of Massachusetts had 14 percent.

“This is shaping up to be a race between three candidates who all cannot figure out how to get past each other, and then everyone else who [is] really vying for fourth place or just survival,” said David Paleologos, director of the Suffolk University Political Research Center, who conducted the survey.

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In fourth place is Senator Kamala Harris of California at 8 percent, followed by South Bend, Ind., Mayor Pete Buttigieg at 6 percent and Representative Tulsi Gabbard of Hawaii at 3 percent.

Among the reasons why Biden, Sanders, and Warren will be difficult to topple from the top tier: a significant portion of their supporters say they have made up their minds about the race.

This is especially the case with Sanders. Nearly half — 48 percent — of his supporters said they would definitely vote for him, including Douglas
Deaett, a 70-year-old retired emergency room doctor from Hanover.

“Bernie has been dedicated to his polices for a long time,” said Deaett. “Most of the other contenders are riding his coattails and wouldn’t work as hard to bring his ideas to fruition.”

Bernie Sanders.
Bernie Sanders.Scott Olson/Getty Images/Getty Images

Some 45 percent of Biden’s supporters say they have found their candidate, while 35 percent of Warren’s supporters consider themselves committed, according to the poll.

There are signs in the poll, however, that Warren might have the most room to grow in the next six months. She was by far the most popular second choice of respondents, with 21 percent saying they would back her after their first choice.

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“This is an especially important number now that a number of candidates could drop out soon, and shows how she could grow,” Paleologos said.

Elizabeth Warren.
Elizabeth Warren.John Locher/Associated Press/Associated Press

Another key number for Warren: 60 percent of voters say they are open to changing their mind and might look to their second choice.

Just over one-fifth of likely New Hampshire Democratic primary voters said they weren’t leaning toward any of the 25 candidates in the survey. Seven of the candidates, including Massachusetts Representative Seth Moulton, of Salem, didn’t have a single respondent say they were backing them.

The survey of 500 New Hampshire voters who said they were either “somewhat likely” or “very likely” to vote in the state’s Democratic primary next year was taken Thursday though Sunday. The poll’s margin of error was plus or minus 4.4 percentage points.

One candidate who may be starting to grab the attention of New Hampshire Democrats is Gabbard, who had just 1 percent in the last Suffolk/Globe poll of New Hampshire, taken in April. In the latest survey, the representative from Hawaii moved into sixth place — ahead of New Jersey Senator Cory Booker, who had just 1 percent support — although the difference is still well within the poll’s margin of error.

But the poll might disappoint the campaigns of Beto O’Rourke, a former representative, and Julian Castro, a former US housing secretary, who both had levels of support that rounded down to zero percent.

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“As far as Beto is concerned this result isn’t surprising given how poorly his year has gone, but considering how he came into the race with such momentum it is still a little stunning to see,” Paleologos said. O’Rourke has been in headlines in recent days because of the mass shooting in his hometown of El Paso, though, given the timing, that likely would not have registered in the poll.

And despite an aggressive television and social media ad campaign, California billionaire Tom Steyer was at 1 percent when his support was rounded up.

Another point that stood out in the poll? New Hampshire voters said they are ready for the historically large field to shrink. Nearly two out of three voters said they felt if a candidate didn’t qualify for the third round of debates they should drop out.

Due to thresholds set by the Democratic National Committee, it’s possible that half the field will not qualify for the September debates. To secure a spot on the stage, a candidate must have at least 130,000 individual donors and reach 2 percent in four sanctioned polls (The Suffolk/Globe New Hampshire poll is not one of them).

On the issues, 85 percent of New Hampshire Democratic primary voters said they want their party’s nominee to back Medicare for All. A majority also want the nominee to back the Green New Deal and impeach President Trump.

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A majority of respondents also confirmed their top priority is a candidate who can beat Trump — even if it means that person doesn’t share their values.

Jennifer Pitts, 54, an executive recruiter for a business consulting firm, is among the 58 percent polled who said the most important thing for Democrats is to nominate a candidate who can stop Trump.

At this moment, she is backing Biden.

“I want to beat Trump, and it pains me to say that I fear only a well-known white guy like Biden can do it,” Pitts said. “But, like New Hampshire voters do, I will be watching to see what happens as the campaign plays out.”


James Pindell can be reached at james.pindell@globe.com. Follow him on Twitter @jamespindell or subscribe to his Ground Game newsletter on politics: http://pages.email.bostonglobe.com/GroundGameSignUp