To the Sanders campaign, his support among non-Democrats is an argument in favor of his electability: Only he can excite the younger, less politically engaged cohort of people who aren’t registered with either party and might otherwise stay home or vote for a third-party contender in November. But the same dynamic could resurrect old resentments within the Democratic establishment that a candidate who for decades has refused to run on the party’s banner in Vermont is poised to capture its presidential nomination.

“You know, he’s not registered as a Democrat, to the best of my knowledge,” former Vice President Joe Biden said of Sanders last week. For years Sanders has resisted entreaties to run as a Democrat in Vermont, though he caucuses with the Democrats as an independent in the Senate.

“Can he truly build a majority coalition within the party?” asked Dante Scala, a political scientist at the University of New Hampshire. The tension could be magnified should Sanders enter the Milwaukee convention with a plurality, but not a majority, of delegates that hail disproportionately from states that allow non-Democrats to vote. “I can imagine the response being perhaps, ‘Well, they weren’t real Democrats,’” Scala said.

Read: What if the parties didn’t run primaries?

First, however, Sanders must focus on New Hampshire, where he faces more competition for undeclared voters than he did the last time around, and his advantage is narrower. Fresh off his near-victory in Iowa, Pete Buttigieg, the former mayor of South Bend, Indiana, is gaining in the polls, and he has made a concerted effort to woo independents and what he calls “future former Republicans.” Buttigieg polled even with Sanders among both Democrats and undeclared voters in a survey released Friday morning and overtook him in another poll over the weekend. Former Vice President Joe Biden is also targeting middle-of-the-road voters and recently touted endorsements from 100 New Hampshire independent.

Yet not all independents are the same: Only about half of them in New Hampshire are truly swing voters, while the rest lean strongly to one side or the other, says Liam Kerr, a co-founder of The Welcome Party, a center-left group operating in New Hampshire and South Carolina. The organization’s goal is to entice undeclared voters who typically vote only in general elections to participate in the Democratic primary.

“There are different types of independent voters, and people are speaking in different ways to them,” Kerr told me. Buttigieg and Biden are going after the traditional centrists who vote for both Democrats and Republicans depending on the election and who’s on the ballot. New Hampshire is likely to have more of these voters up for grabs this year. Undeclared voters can choose to vote in either party primary, and while they split between the ultra-competitive Democratic and Republican contests in 2016, only the Democratic primary is a real toss-up on Tuesday.