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British Conservative MP Penny Mordaunt speaks at an event to launch her campaign to be the next Conservative leader and Prime Minister, in London, on July 13.HENRY NICHOLLS/Reuters

The list of candidates vying to succeed Boris Johnson as leader of Britain’s Conservative party is rapidly narrowing, with party members on course to select either the country’s first racialized prime minister or the third woman to lead the country.

After a second round of voting by Conservative MPs Thursday, former chancellor of the exchequer Rishi Sunak and junior trade minister Penny Mordaunt solidified their positions as the top two candidates.

Only five candidates remain, and Tory MPs will continue voting next week until just two are left. The party’s roughly 200,000 members will then select the winner through a national ballot next month, and the result will be announced Sept. 5. Whoever wins will take over from Mr. Johnson as both party leader and prime minister.

A look at the five U.K. Conservative MPs vying for the leadership

Mr. Sunak took 101 votes Thursday, up 13 from Wednesday’s first round. Ms. Mordaunt came second with 83 votes, 16 more than she received in the first ballot. Foreign Secretary Liz Truss finished third with 64, up 14 from Wednesday.

Ms. Mordaunt’s campaign has been surging since she announced her bid Sunday. Several polls of party members have put her well ahead of Mr. Sunak and Ms. Truss, and if she can secure one of the final two spots, most pundits expect her to win.

However, Ms. Truss has been trying to position herself as the heir apparent to Mr. Johnson, who resigned last Thursday, and the best candidate to defeat Mr. Sunak.

Mr. Johnson’s supporters are believed to be furious at Mr. Sunak, 42, for resigning from cabinet last Tuesday and then leading the calls for Mr. Johnson to go. Media reports say they are privately urging MPs to back Ms. Truss, who did not resign her post. “I’m a loyal person, I’m loyal to Boris Johnson,” she told reporters Thursday.

She has also vowed to cut taxes in an apparent shot at Mr. Sunak. He has warned Tories not to be swayed by “fairytale” promises of lower taxes while government spending remains high. Instead, he has vowed to cut taxes only when public finances have improved.

Ms. Truss, 46, has said that taxes should be slashed right away. “Under my leadership, I would start cutting taxes from day one to take immediate action to help people deal with the cost of living,” she wrote in The Daily Telegraph this week.

Ms. Mordaunt, 49, has tried to claim the centre ground. While she was a vocal supporter of Britain leaving the European Union during the 2016 Brexit referendum, she has backed several progressive social causes, including transgender rights. She has also been less strident on tax cuts than Ms. Truss. “I am a small-state, low-tax conservative, but I also believe we need to use the levers of government to support jobs and livelihoods through difficult economic situations,” Ms. Mordaunt wrote in the Telegraph.

The next round of voting among MPs takes place Monday, and the candidates will participate in a televised debate on Sunday.

As it stands, party members will most likely be selecting either a prime minister whose family hails from India or the third woman to lead the party and the country, after Margaret Thatcher and Theresa May.

Even the candidate who finished fourth in Thursday’s vote, Kemi Badenoch, would be a first as prime minister if she made it to the final round.

Ms. Badenoch’s parents are from Nigeria, and she spent several years in Africa as a child. She worked at McDonald’s as a high school student in London and went on to earn university degrees in engineering and law.

She won the support of 49 MPs Thursday and is considered a rising star among many right-wing Tories. However, it’s widely expected that Ms. Badenoch, 42, will face pressure to end her candidacy and support Ms. Truss.

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