An initial field of 11 challengers has been whittled down following two days of votes by lawmakers from the ruling Conservative Party. But no individual has yet emerged as the obvious successor to Boris Johnson who announced he was stepping down following a series of scandals.
While Sunak has topped those two votes, he faces stiff competition from Truss, who has the backing of a number of senior figures, and junior trade minister Penny Mordaunt, who polls suggest is the most popular with party members who will decide the winner.
Former equalities minister Kemi Badenoch and Tom Tugendhat, chair of parliament's Foreign Affairs Committee, also remain in the running but trail the others in support from Conservative lawmakers.
A snap poll by market research company Opinium of the British public -- who do not get a say in the Conservative Party decision on the next prime minister -- showed Tugendhat was viewed as the best performer by 36% of viewers.
Sunak came second on 24%, followed by Mordaunt and Badenoch on 12% and Truss at the rear on 7%.
Whoever gets the job will take on rocketing inflation and low economic growth, as well as the public's lack of confidence in politics after Johnson's scandal-ridden time in power.
Sunak and Truss tussled over economic policy in the debate, hosted by broadcaster Channel 4.
Truss has proposed scrapping increases in payroll tax and corporation tax proposed by Sunak, at a cost of over 30 billion pounds ($36 billion) a year, to be funded by a slower reduction in government debt built up during the COVID-19 pandemic.
"We have to be honest, borrowing your way out of inflation isn't a plan, it's a fairy tale," Sunak told Truss.
Truss said tax rises would undermine business investment just as the economy was faltering.
"You cannot tax your way to grace," she said.
Voter polls also suggest the Conservatives are falling significantly behind the opposition Labour Party.
"I'm very aware that while my party chooses a new leader, you are watching us pick your next prime minister... I hope you like at least one of us," Mordaunt told television viewers.
Tugendhat got applause from the studio audience for most clearly distancing himself from Johnson -- shaking his head when asked if he trusted the prime minister -- while Badenoch said her rival had dodged tough decisions by never serving in Johnson's government.
Sunak, whose decision to quit the Treasury last week helped trigger a cascade of ministerial resignations that brought down Johnson, remains the favourite among his 358 Conservative parliamentary colleagues.
But his lead over Truss and Mordaunt is slim, and both could overtake him depending on whom lawmakers who backed other candidates choose to support. The battle has become increasingly hostile as the rivals fight to stay in the contest.
On Thursday, Attorney General Suella Braverman was knocked out of the race and she has thrown her support behind Truss, who has also gained the backing from David Frost who negotiated Britain's exit from the European Union.
The Times newspaper reported that Johnson was urging defeated leadership candidates to back "anyone but Rishi".
Meanwhile Mordaunt, a less well-known figure among the public at large who has become the bookmakers' favourite, is facing increasing attacks from rival camps over her experience, with Frost saying she was not tough enough towards the EU, a key issue for many Conservatives.
Both Truss and Badenoch also criticised Mordaunt for taking what they viewed as too liberal a stance on transgender issues.
Ballots of lawmakers will resume on Monday with the candidate with the fewest votes eliminated each time until a final two are chosen by July 21.
The new leader will then be selected by the country's 200,000 Conservative Party members, and will be announced on Sept. 5.
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