Two hours later, the remaining candidates will then do battle in a BBC televised debate, following a first TV showdown on Sunday that Johnson skipped.
He said he believed he could negotiate a new deal with the European Union which would avoid the need for the Northern Ireland backstop, which led to Theresa May's deal being defeated three time in the Commons. While he has vowed to respect the October 29th Brexit day while others - including Michael Gove - have said they would consider delaying Brexit again, Nigel Farage has raised doubts about Johnson's dedication and reliability.
Foreign Secretary Jeremy Hunt said that committing to a "hard stop" at the end of October would be to ensure a no-deal Brexit.
Mr Johnson gained the support of former leadership contender Andrea Leadsom on Tuesday morning, who said he was the "best placed to get us out of the European Union at the end of October" and an "election winner".
The British currency had dropped to its weakest since January against the euro and dollar earlier in the session, as investors worry that Johnson could put Britain on a path towards a disruptive no-deal Brexit.
Mr Johnson's bid has been spearheaded by South Staffordshire MP Gavin Williamson, who maintains he is the only candidate who can deliver Brexit and defeat Jeremy Corbyn.
But Stewart said a no-deal Brexit would be "deeply damaging" and threatening self-harm is an attempt to extract concessions from Brussels was "nonsense".
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Johnson has committed to the upcoming BBC debate, only his second public appearance in connection to his leadership campaign. We can stop a no-deal Brexit much more easily than that.
Ahead of the second round of voting on Tuesday, Mr Raab hit out at those in Parliament who he accused of trying to "steal" Brexit from the voters.
Environment minister Michael Gove also took a dig at Johnson, saying his absence from Monday's hustings in Westminster was described to him as being like "Hamlet without a prince". Home Secretary Sajid Javid joined in saying: "You don't deliver on democracy by trashing democracy".
Mr Johnson is snubbing the debate, claiming the proposed six-way programme would be "cacophonous" and "blue-on-blue", with "loads of candidates".
The presenter, Krishnan Guru-Murthy, also challenged them on more controversial issues that have come up in the campaign such as Gove's past cocaine use or Raab's critique of feminism and Stewart's repeated U-turns.
Candidates who do not receive the backing of more than 33 of the 313 Conservative lawmakers in today's vote will be eliminated.
The second ballot results of the race for Conservative leadership sees Boris Johnson leading. Johnson, the former foreign secretary and London mayor, skipped the first debate on June 16. The mainly pro-Brexit Conservative Party members, some 160,000, will cast the deciding votes in a weeks-long process that will culminate in the announcement of the victor on July 22. Raab got 27, Javid got 23 and Stewart was on 19.
The result of Tuesday's second round of voting is due sometime after 1 p.m. ET., and those left standing will participate in a BBC debate tonight.