Stephen Booth is Director of Policy and Research at Open Europe.

The only issue that really matters in the Conservative leadership contest is Brexit. It is impossible to predict what will happen because there is no silver bullet available to the new Prime Minister, who will remain at the mercy of events in Westminster and Brussels.

The European elections demonstrated that there isn’t a stable majority for either leave or remain, and it’s still the case that the impasse can only be resolved in one of three ways: No Deal, negotiated exit or Remain. Sam Gyimah, who has pulled out of the race due to insufficient support, was the only potential candidate offering, via a second referendum, the Remain option. The debate now is whether the priority should be leaving the EU by October 31st , with or without a deal, or getting a Brexit deal ratified by Parliament. Some advocates of a negotiated exit appear to be completely ruling out leaving without a deal, others have not.

While these are important dividing lines, the most significant question for our would-be Prime Ministers is not “What?” but “How?”. It has been noted elsewhere on this site that the new Prime Minister is no more likely to be in control of events than Theresa May and will have to contend with similar roadblocks in both Westminster and the EU.

First, it is extremely unlikely that it will be in the Prime Minister’s gift to ensure that we exit on October 31st  in the absence of a deal. This House of Commons, aided by its Speaker, appears no less determined to prevent this outcome than it has to date. It has already successfully forced extension after extension and a no confidence vote is clearly conceivable. Second, it is just as unlikely that a new leader can successfully steer the current Brexit deal through Parliament, merely by force of new personality. The deal’s opponents are too dug in.

These immovable obstacles seemingly point in only one direction: a general election. Indeed, insisting on a hard deadline could precipitate one. But neither is this much of a plan. Realistically, only a sizeable Commons majority would break the deadlock. Given the parlous state of the Conservatives’ poll ratings, the rise of the Brexit Party and the resurgence of the Liberal Democrats, an election would be an immense gamble to take, from which the Party might struggle to recover.

So, the likelihood is that, one way or another, the EU will have a big influence on how Brexit plays out from here to 31st October and, potentially, beyond.

Ironically, the individual with the most power to force a No Deal exit could be the French President. Emmanuel Macron had misgivings about granting the UK a long extension at the last time of asking and he might, might just be prepared to veto another or set a timetable, effectively forcing the UK to make a choice.

Implicitly, this appears to be the hope of some candidates. This choice might include a deal or be between No Deal and revoking Article 50. However, it appears the rest of the EU really doesn’t have the stomach for No Deal and would baulk at imposing a firm deadline on the UK. So, on balance, the EU is more likely to kick the can, but this cannot be taken for granted.

Meanwhile, before we get to October, a new Prime Minister is bound to attempt to revisit the Brexit deal. The EU is sticking to the script that the deal cannot be reopened and that the negotiations have finished. But, unless it is seriously prepared to entertain No Deal, the EU will have to engage, however reluctantly, with demands to renegotiate the current package. After all, it was the failure to pass the existing deal which cost May her premiership.

Rewriting the Political Declaration with greater emphasis on “alternative arrangements” would be relatively easy but is unlikely to cut it in Westminster. There is growing media speculation, from those on the ground in Brussels, that there could be movement on the backstop. There is talk of a long time-limit or a firm timetable to phase out elements of it. This is not entirely inconceivable, but one would expect this to be at the very outer limits of what the EU might consider.

However, the challenge for any leader in delivering a new package will not only be to tread carefully in Brussels, and above all Dublin, but to find the numbers in Parliament to back a deal. It probably means getting the Democratic Unionist Party back on board because the Tory No Deal faction could well increase after this contest. Securing the votes of Labour rebels has proven fruitless so far but might be helped if Jeremy Corbyn finally embraces a second referendum. The point is that the space to land a deal now appears to be vanishingly small.

Ultimately, every candidate’s Plan A requires defying the odds. It would be brave to bet against further drift.