Tory leadership rivals Boris Johnson and Jeremy Hunt are at loggerheads over how the UK should leave the EU.
Both contenders for prime minister claim they can renegotiate a Brexit deal that the EU says is closed.
Mr Johnson said the UK must leave on 31 October “deal or no deal” but Mr Hunt called this a “fake deadline” that could trigger a general election if Parliament rejects a no-deal Brexit.
The winner of the contest will take over from Theresa May on 24 July.
In an interview with Talk Radio, Mr Johnson insisted he would take the UK out of the EU by Halloween “come what may, do or die” and has challenged his opponent to make the same commitment.
Mr Hunt said he would leave the EU without a deal, but not if there was a “prospect of a better deal”.
It comes as International Trade Secretary Liam Fox – who is backing Mr Hunt – again criticised Mr Johnson’s claim that the UK could continue tariff-free trade with the EU in the event of a no-deal Brexit.
Mr Johnson has argued that a provision under the General Agreement on Tariffs and Trade – known as Gatt 24 – could be used to avoid tariffs for up to 10 years.
But Dr Fox said this would require the agreement of the EU, which Brussels has made clear it would not give.
He added it was important that public debate on the topic was conducted “on the basis of fact rather than supposition”.
On Tuesday, Mr Johnson conceded that his plan would require the approval of the rest of the EU, but insisted it was still “an option”.
Why EU is unfazed by no-deal threats
Boris Johnson and Jeremy Hunt say they are serious about pushing for a no-deal Brexit if they are unable to negotiate a better withdrawal agreement with Brussels.
Yet the EU seems unfazed. Why, when we know EU leaders want to avoid a no-deal Brexit?
Part of the reason, at least, is time.
It’s summer. European capitals are sweltering under a heatwave with government ministers counting the days until they hit the beach or find some cool mountain air.
The day the Brexit extension runs out – 31 October – seems an eternity away in political terms.
Also, just as Messrs Johnson and Hunt do not accept the EU’s word when it says the Withdrawal Agreement cannot and will not be re-negotiated, EU leaders do not take them at their word when they threaten no deal by the end of October.
Meanwhile, Mr Johnson has recruited former Tory leader and Brexiteer Iain Duncan Smith as his campaign chairman, in an attempt to boost his leadership bid, the Telegraph reports.
It comes as the former foreign secretary continues to face scrutiny about Friday’s row with his girlfriend Carrie Symonds, which prompted neighbours to call the police.
After days of criticism that he was hiding away, Mr Johnson undertook a series of media and public appearances on Tuesday – but declined to answer questions about the argument.
On Tuesday Mr Hunt told the BBC the next prime minister should be someone who is “trustworthy” and the ability to negotiate a new Brexit deal was about “personality”, but stopped short of directly criticising his rival.
The day before, Mr Johnson told the BBC anyone questioning his character was “talking absolute nonsense”.
The two remaining leadership hopefuls will face more questions from the public on Wednesday in a digital hustings, streamed on the Conservative Party’s Facebook and Twitter accounts.