The Irish border was the conundrum Theresa May couldn’t crack.

Her solution, the Irish backdrop - an insurance policy to ensure the border remained free to cross of checks in the event of no deal between the UK and the EU - proved popular among Northern Ireland business but not with Northern Ireland’s DUP in Westminster.

The Democratic Unionist Party’s 10 MPs hated it and voted down the Withdrawal Agreement down three times as a result.

It may have ensured no hard border between Northern Ireland and the Republic, but it did so by giving the North special status with certain EU regulations applying only to NI and not the rest of the UK, with no time limit and no legal means of leaving without the EU’s permission.

Boris Johnson with DUP's deputy leader Nigel Dodds and party leader Arlene Foster. Credit: PA

The DUP saw that as a threat to the very existence of the United Kingdom and Northern Ireland’s place in it.

With the backing of sizeable chunk of Tory Brexiters, their opposition to the backstop buried the Brexit deal agreed between Britain and the EU, and with it Mrs May’s premiership.

The men vying to replace her are vowing to replace the backstop with alternative arrangements.

Jeremy Hunt and Boris Johnson will face some of Northern Ireland’s 500 plus party members at the latest leadership hustings on Tuesday in Belfast, and the question of what exactly those alternative arrangements will be, and how they will guarantee an open border between north and south, will undoubtedly come up.

The Irish border. Credit: PA

Mr Hunt has said he would pursue a "technology-led solution" to the Irish border issue, while Mr Johnson has claimed it can be resolved by applying "technical solutions".

David Haldane runs a builders merchants in Newry, three miles from the Irish border. He tells me he has been unimpressed by the promises of the two Tory leadership contenders.

"I am deeply worried. We can not afford a hard border, some of our biggest customers are already starting to take their business elsewhere in preparation for problems at the border. Our staff live on both sides of the border.

"I see a lot of political posturing, but no genuine solutions to keep the border open."

Mr Haldane praised the Irish backstop solution, which he said ensured a fluid border and allowed trade to continue between the UK and the EU.

Theresa May speaking in Northern Ireland last November. Credit: PA

Three miles south, we spoke to one man who works on one of the 208 border crossings which divide Northern Ireland from the Republic, just off the Dublin Road.

He works on the UK side of an unmarked, totally invisible line in the middle of a tiny bridge running over a small river.

"I cross the border 4 times just to go home for my tea everyday,” he says.

"Brexit is a disaster. Border posts, check points, are impossible. Who’s going to pay for them?"

Both Mr Johnson and Mr Hunt say they will not countenance a hard border, but both are pledging to leave the EU on October 31 without a deal if necessary - that risks a hard border, whether they want one or not.

Johnson and Hunt will go head-to-head at the latest Tory hustings event in Northern Ireland. Credit: PA

Both candidates will face questions from a tiny portion of the party’s membership on Tuesday.

Their answers on the specifics of what exactly their alternatives are to Theresa May’s failed backstop will ultimately have little bearing on the result of this contest.

A recent YouGov poll of Tory members found 59% would prioritise leaving the EU even if it meant Northern Ireland breaking away from the rest of the UK.

But whoever convinces those Tory members to make them prime minister will then have to convince the EU of their alternative plans, and then convince Parliament to pass it.

The Irish Border remains the unsolved conundrum haunting the Conservative Party.