Pitching himself as the candidate to offer a clean start for the Conservative Party, backbench MP Mr Tugendhat was widely seen to have performed strongly in the campaign and TV debates but was knocked out of the contest in the third ballot of Tory MPs after receving just 31 votes.
All eyes will already turn to the next ballot on Tuesday, with Kemi Badenoch now the candidate with the lowest share of the vote among the four candidates that remain.
Mr Sunak, who got 115 votes, edged closer to 120 votes required to guarantee a place in the final two, who will face a vote of the Tory membership to decide the next party leader and prime minister.
It remains unclear which of the three remaining candidates will join Mr Sunak in the final two.
After his elimination from the leadership contest, Mr Tugendhat said he will listen to what other candidates have to say before deciding who to support.
In a video posted to Twitter, Mr Tugendhat said: “That is the end of the road for me in this race, but look, it has been an amazing run, I’m incredibly proud of the team, I’m incredibly grateful to all the supporters who have been with me at some - if not all - stages of the race we fought for a clean start because we know that that’s what the country is crying out for.”
Ms Mordaunt stayed in second place despite a difficult few days which saw her attacked by all sides with allegations over her ministerial record and questions about her previous statements on trans rights.
Both Ms Truss and Ms Mordaunt have also had their economic proposals blasted by the former chancellor Mr Sunak in recent days, who lashed out at the rival candidates’ tax plans during the TV debate on ITV on Sunday night.
In a sign of the concern about the way the leadership race is being conducted, Mr Sunak and Ms Truss confirmed they did not want to take part in a Sky News debate planned for Tuesday – prompting the broadcaster to cancel the show.
Ms Badenoch, who saw her vote jump from 49 to 58 in the latest round, said she had the “momentum” behind her.
She tweeted on Monday evening: “Continued momentum, closing the gap, I am the only change candidate left in the race. I’m in it to win.”
If she is eliminated next, the votes of Badenoch backers could determine who faces Mr Sunak in the vote of Tory members.
An MP supporting Ms Truss insisted the Foreign Secretary was “heading in the right direction”.
Asked about the increase in support for Ms Badenoch, they said: “Liz is absolutely heading in the direction… she’s ahead and she’s heading in the right direction, so that’s good.”
Attorney General and right-winger Suella Braverman, who was knocked out of the contest last week, had thrown her weight behind Ms Truss.
That endorsement, however, appears to have not been enough to push the Foreign Secretary into second place.
Ms Mordaunt will continue into Tuesday knowing that her place behind Mr Sunak remains vulnerable.
After the ballot, she tweeted: “My vote is steady and I’m grateful to my colleagues for all their support and thrilled to be in second place once more.
“MPs know that I’m a strong candidate, running a truly clean campaign and putting forward a positive vision for the party and our country.”
On Monday, Ms Mordaunt faced accusations that she missed ministerial meetings because she was plotting her Tory leadership bid.
The trade minister’s absence from meetings forced colleagues to pick up the pieces, International Trade Secretary Anne-Marie Trevelyan earlier alleged.
International Trade Secretary Anne-Marie Trevelyan says there have been 'a number of times' where Penny Mordaunt 'hasn't been available and other ministers have had to pick up the pieces' in recent months.#CallTheCabinet | @annietrev | @NickFerrariLBC pic.twitter.com/aYdkoO1JGj
— LBC (@LBC) July 18, 2022
Asked about Ms Mordaunt’s grasp of details, Ms Trevelyan said: “We all do our jobs in different ways.
“Understandably, perhaps, now it’s clear, Penny has for the last few months spent some of her time focused on preparing her leadership campaign, for which I have utmost respect, that’s how this system works.”
Ms Trevelyan added: “There have been a number of times when she hasn’t been available, which would have been useful, and other ministers have picked up the pieces.”
But former minister Harriet Baldwin, an ally of Ms Mordaunt, said: “Having worked with Penny Mordaunt for a number of years, I can confirm this is not true.
“She is one of the most hard-working ministers and constituency MPs I know. I look forward to Penny continuing to put forward her positive vision for Britain.”
— Penny Mordaunt (@PennyMordaunt) July 18, 2022
Cabinet minister Kit Malthouse said he expects the Tories to pull together in a “spirit of harmony and love” after the leadership battle.
“All political parties are standing coalitions and the Conservative Party is the same,” he told Sky News.
“A vigorous exchange of ideas, in what is a challenging time for the country, should be expected when you are talking about such important issues and the leadership of a G7 nation.
“If it was just a polite agreement and consensus across the board, there wouldn’t be much point in having a competition at all.”
Mr Malthouse has not publicly declared his support for any of the candidates to succeed Mr Johnson.
A series of votes among Tory MPs this week will narrow the field down to a final two, who will then face a summer of campaigning for the support of party members in a final vote.
The new leader will be announced on September 5 and is expected to become prime minister the following day.
Mr Sunak said that if he is successful his first foreign trip will be to Kyiv to stress the UK’s continued support for Ukraine in its fight against the Russian invasion.
He told The Sun: “I will reinforce our policy of total support for Ukraine that Boris has so ably led.”
But Ms Mordaunt said her campaign has the support of 10 Ukrainian MPs.
Meanwhile, Ms Truss seized on analysis by the Centre for Economics and Business Research which suggested that tax revenues in 2024/25 will be around £60 billion more than the Office for Budget Responsibility estimates, partly due to the impact of high inflation.
A spokeswoman for Ms Truss said: “The CEBR analysis shows that there is money for tax cuts whilst still bringing debt down.”