When he accepted the Libertarian Party’s nomination in May, Gary Johnson told delegates that he was ready to peel away votes from the Republican and Democratic candidates, potentially catapulting a party on the fringe of U.S. politics onto the presidential debate stage and, with luck, to the White House.

But nearly four months later, Johnson’s presidential chances appear shaky. The former Republican governor of New Mexico is polling around 10 percent nationally, healthy for a third-party candidate, but below the 15 percent threshold required to be included in presidential debates.

And Thursday, Johnson’s credibility suffered a blow when he fumbled over a question about the crisis in Syria, asking an MSNBC commentator “What is Aleppo?”

The challenges facing Johnson’s bid go beyond a mental short-circuit and reflect the deeply entrenched obstacles that third-party and independent candidates face when running for president.

On the surface, the low approval ratings for Hillary Clinton and Donald Trump present a prime opportunity for a candidate like Johnson, who is socially liberal and fiscally conservative. Even so, that intriguing blend of policies has made it difficult for the Libertarian ticket, which includes William F. Weld, the former Republican governor of Massachusetts, to attract stray Democrats or disenchanted Republicans in large numbers.

“He’s had issues coalescing the anti-Trump Republican crowd, partially because it’s a mix of social conservatives and moderates, and partly because at times he’s seemed more keen on appealing to the Bernie bros,” said Tim Miller, a Republican and a former aide to Jeb Bush, referring to the supporters of Sen. Bernie Sanders of Vermont.

Miller opposes Trump and is considering voting for the Libertarian ticket this year.

Miller added that Johnson’s flub about Aleppo did not make him a riskier bet on foreign policy matters than is Trump. But, he said, it does highlight the problem that many Republicans have with Libertarians.

“It reinforces my top policy difference with him, which is his relative isolationism” on foreign affairs, he said.

While prominent Republicans often speak highly of Johnson and Weld, few big names have come forward to formally endorse him. Johnson drew criticism from some Republicans last week when he took issue with the expression “illegal immigrant,” and his views on fully legalizing marijuana remain out of step with mainstream politics.

Still, Johnson’s supporters have expressed hope that anti-Trump sentiment could help propel the Libertarians this year.

On social media, many of Johnson’s backers say they hope that Sen. Jeff Flake, the Arizona Republican who has had a public feud with Trump, might get behind the Libertarian ticket, but so far they have had no luck converting him. In June, Sen. Ben Sasse of Nebraska expressed interest in the Libertarian Party but later said that Johnson’s positions on abortion, religious liberty and national security were deal breakers.

Mitt Romney, the 2012 Republican nominee who has showered praise on Weld, pushed publicly this week for the Libertarians to be allowed in the debates. Still, he has given no sign that he will back the party’s ticket in November, even though he has said he will not vote for Trump or Clinton.

“The Libertarians just break too much from traditional GOP orthodoxy on foreign policy and social issues to get a big amount of Republican support,” said Kyle D. Kondik of the Center for Politics at the University of Virginia.

The Democratic electorate has also failed to provide fertile ground for the Libertarians, despite the lingering disappointment among some who supported Sanders. Jill Stein, the Green Party nominee, has drawn some Sanders backers, and Johnson’s position in favor of free trade is out of step with most Democrats these days.

Political analysts point to a variety of reasons third-party candidacies tend to flounder in the United States. A lack of media exposure is often cited, as are political science theories that argue that in a winner-take-all political system, voters gravitate to two mainstream parties out of fear that they would otherwise be wasting their votes.

Many see the debates as the biggest obstacle for third-party candidates. The Commission on Presidential Debates requires that candidates invited to the three nationally televised presidential debates have an average of at least 15 percent support in five specific national polls. Johnson has said that he has no chance of winning the election without debating Trump and Clinton, and his supporters argue that the poll threshold is arbitrary and too high.

“In the real world, that’s just impossible,” said Richard Winger, an expert on third-party candidates and the editor of Ballot Access News. “That’s a real barrier.”

Stuart Stevens, a Republican strategist who has worked with Romney and Weld, said that difficulties raising money for campaign infrastructure and advertising put the Libertarians at a distinct disadvantage in reaching voters who might gravitate to their message.

“Most people don’t know who Gary Johnson is,” said Stevens, adding that he remained open to voting for the candidate despite his slim chances. “He doesn’t have much of an apparatus to push him.”

Despite the hurdles, Johnson could still have a major effect on the race.

Polls show him appealing to young, educated voters in states like North Carolina who might otherwise support Clinton, and disaffected conservatives in places like Utah, where much of the traditionally Republican Mormon population is turned off by Trump. In a close election, a Libertarian ticket that is drawing one-tenth of the vote could reshape the electoral map and tilt battlegrounds in unpredictable directions.

A Quinnipiac University survey released Thursday showed Johnson pulling a big chunk of the vote in Ohio, highlighting the influence that he might have.

“Gary Johnson could decide the presidential election in the Buckeye State,” said Peter A. Brown, assistant director of the Quinnipiac University poll. “He is getting 14 percent from Ohio voters, and how that cohort eventually votes could be critical in this swing state — and in the nation.”