Tom Tugendhat has become the latest candidate to be knocked out of the Conservative leadership race.
In the third round of voting by Tory MPs, the contenders received the following votes:
Kemi Badenoch, 58
Penny Mordaunt, 82
Rishi Sunak, 115
Liz Truss, 71
Mr Tugendhat, 31
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Two more votes are due to take place on Tuesday and Wednesday until two candidates remain.
They will then face a summer of campaigning and hustings before a vote by the wider party membership, with the winner expected to be announced on 5 September.
The remaining runners will not face each other in a debate that had been planned by Sky News for Tuesday after Mr Sunak and Ms Truss refused to take part.
Conservative MPs are said to be concerned about the damage the previous debates have done to the image of the party after exposing disagreements and splits between the leading candidates.
Labour leader Sir Keir Starmer said he was "astonished that those that want to be prime minister of the United Kingdom are pulling out of debates and out of scrutiny".
Exchanges between the candidates have been feisty during the two TV debates that have taken place, with rows focused over their plans for tax.
Although Mr Sunak, the former chancellor, has insisted now is not the time to reduce the tax burden, others have said they would take immediate action.
In the ITV leadership debate on Sunday, Mr Sunak accused Ms Truss of peddling "something-for-nothing" economics after she said he would "choke off growth" by raising taxes to their highest level in 70 years.
There were further furious exchanges between Ms Mordaunt and Ms Badenoch on Friday's Channel 4 debate over the issue of self-identification for trans people.
The contest was triggered after Boris Johnson announced he would be leaving Downing Street after an avalanche of ministerial resignations over his handling of the Chris Pincher scandal.
He will remain as prime minister until his successor has been named.
Mr Johnson has come under scrutiny this week for not attending three emergency COBRA meetings about the heatwave.
A Number 10 source said the prime minister had been in contact with relevant ministers all weekend.
Kit Malthouse, the chancellor of the Duchy of Lancaster, who chaired the meetings, insisted Mr Johnson was "right across" the situation.
Lisa Nandy, the shadow communities secretary, said the prime minister has "clearly clocked off" since announcing his resignation, urging him and all his ministers to "turn up for work".