The race to succeed Theresa May as British Prime Minister is entering its crucial stage. In the next two days, Conservative lawmakers will select two candidates to be paraded in front of the wider party membership for a final decision on who will take on the fateful (and twice fatal) Brexit chalice.

But something strange has happened. All the attention is not on the rockstar front runner, Boris Johnson, but an insurgent outsider.

Rory Stewart, the current international development secretary, has been the hands-down star of this leadership race. Initially scoffed at by Conservative commentators, Stewart, a self-declared centrist who still backs Theresa May's Brexit deal, has seen off five candidates, all of whom stood on much harder Brexit platforms. His campaign has clearly spooked the others, who are seeing some of their supporters defect to Stewart.

Improbably, on Tuesday he made it through to the next round of voting, winning the support of 37 Conservative members of Parliament – up from 19 in the last round. It was the biggest leap forward of Johnson's remaining rivals.

On the face of it, Stewart's campaign looked doomed to fail. He was virtually unknown before the contest began, with barely a month's experience in May's Cabinet and a low-profile job as a junior prisons minister before that. Pitching himself as the only candidate telling the truth about Brexit, Stewart says that May's deal with Brussels is the only one any prime minister can realistically secure. He says that candidates claiming they can take the UK out of the EU without a deal are lying — because Parliament will not realistically allow them to do it.

And he openly mocks his rivals “competition of machismo” claims that simply by “believing in Britain” they can get a better Brexit deal. In a live TV debate on Sunday, he likened their claims to trying to fit too much rubbish in a small bin; to merely “believe in the bin” was not enough, he said.

Stewart is a former diplomat who faces near-constant questions as to whether he worked for British intelligence agencies as a spy. In 2002, he travelled across Afghanistan on foot, a dangerous undertaking at a time when the country was still rife with Taliban. His account of the walk became a best-selling book.

He served as a coalition official in the aftermath of the Iraq war, where he worked resolving tribal disputes among other things.

Stewart's equally unusual campaigning style has been a hit with the public. Over the past few weeks, he has been publishing videos on social media, visiting areas of the country that don't traditionally vote Conservative. Those videos have gone viral.

He says he is a pragmatist who can heal divisions in British society by listening to the public. He claims he is proud of the fact that both Labour and Liberal Democrat voters say they would vote for him.

The Conservative party currently heads a minority government. Boris Johnson is trying to pitch himself as the only true Brexit candidate. He wants a new deal or he will leave with no deal. This sort of language might go down well with ultra-Brexiteers, but it's unlikely to win over many people in the centre.

In contrast, Stewart believes that with his wider appeal, he can win a majority for the Conservative party. He believes that Johnson's Brexit promises are nothing short of a fantasy and would ultimately result in the UK going back to square one with Brussels, or worse, holding a general election that the Conservatives would lose.

However, while he has cemented his position as the “Stop Boris” candidate, he is still only in fourth place among Conservative lawmakers. That might change and he might attract more support as MPs conclude he is the only real alternative to Boris. But after a somewhat lacklustre performance against Johnson in the latest TV debate of the campaign on Tuesday night, he might not.

The other remaining candidates all, like Johnson, claim that they can get a new deal from Brussels and talk about no deal as a possibility. If they knock Stewart out this week, his pragmatic, centrist approach will be dead.

And of course, even if he does make the final two, defeating Johnson is an enormous mountain to climb. Johnson currently has more backers than the next three candidates combined.

Still, the main takeaway from this unusual week in British politics is that halfway through the contest to replace Theresa May, there's no guarantee that Johnson – or another hardline Brexiteer – will get the top job.

As ever with Brexit, everything is fluid.