After several months where the Brexit Party, the Liberal Democrats, Labour and the Conservatives have been locked in essentially a four-way tie in the opinion polls for the next British general election, the election of Boris Johnson as Conservative party leader and prime minister has led to the Tories having a consistent lead over their rivals.

The Tories have a lead of around 10 percentage points over Labour and the Liberal Democrats, who remain essentially tied on around 21%. The Tories’ gains appear to have been made at the expense of the Brexit Party, who have fallen to 14% from previous highs of 26%.

By historical standards, 31% would be a very low Conservative vote share, only barely scraping ahead of their worst ever previous election results in 1997 (30.7%) and 1832 (29.2%). However, the unprecedented fragmentation of party support and the quirks of the first-past-the-post system mean that this result could lead to a substantial majority in the House of Commons.

Despite the inherent issues with trying to forecast how opinion polls might map to results in individual constituencies, to get a sense of how a general election might turn out given current polling, I adjusted the 2017 election results using the latest YouGov poll to capture the movement of voters between parties. Assuming no pact of any sort between the pro-remain parties. The projected totals were:

Conservatives 31% (398 seats)

Labour 22% (142)

SNP 4% (57)

Liberal Democrats 19% (28)

Plaid Cymru 1% (4)

Green 8% (1)

The Speaker, John Bercow, and Claire Wright, an independent candidate in East Devon, would also be projected to win seats.

The movement of seats is broken down in the chart below. The main movement of seats would be from Labour to the Conservatives, with the Liberal Democrats only making modest gains at the expense of the two largest parties.

To illustrate the sort of seats that the Conservatives would expect to gain, I’ve broken out the results for three constituencies with their election results since 2005.

Croydon Central, traditionally a close Labour/Tory marginal in suburban London

Rother Valley, a historically safe Labour seat in South Yorkshire where Ukip came second in 2015

Newport East, a traditionally Labour seat in Wales

In each of these seats, the Conservatives would expect to gain the seat from Labour, even though their vote share is expected to fall, because the Labour vote will drop by more. Neither the Liberal Democrats nor the Greens (and Plaid Cymru in Newport East) are in contention.

However, the Lib Dems, the Greens and Plaid Cymru are expected to announce that they will be forming an electoral pact (“Unite to Remain”) where they will agree one party to stand as a unity anti-Brexit candidate. Might such a pact be effective in stopping the Conservatives from achieving a majority in the House of Commons?

I ran the model again, but combined the Lib Dem, Green and Plaid Cymru vote shares into one candidate. This is being generous – there may well be a sizeable element of (say) Green voters who would prefer to vote Labour or abstain if there is no Green Party candidate running. These concerns aside, the results would be expected to be as follows.

Conservatives 31% (382 seats)

Labour 22% (98)

“Unite to Remain” 28% (95)

SNP 4% (55)

Even if the Lib Dems, Greens and Plaid were able to perfectly combine their vote, it would barely make a dent in the Conservatives’ majority as the bulk of their gains would be a Labour’s expense. The results are summarised in the following chart (remain alliance seats are denoted in pink).

To see why denying the Conservatives a majority, consider the following list of 500 Unite to Remain target constituencies, arranged from easiest to most difficult, and coloured by the winner of the 2017 election.

Aside from a handful of Tory seats such as Richmond Park, the vast majority of the Alliance’s target seats are held by Labour and therefore ineffective at denying the Tories a majority.

The charts below show the projected majority for the Unite to Remain alliance, or the amount of the vote that they are expected to fall short of victory. The chart on the top shows the party that won each seat in 2017, whilst the chart on the bottom shows the projected winner if an election was to be held tomorrow.

Should the alliance somehow be able to beat expectations, agree unity candidates in every constituency in Great Britain, and are perfectly able to combine the votes of the three parties, they will still find it difficult to deny the Tories a majority as the bulk of their gains will be at Labour’s expense.

If the Unite to Remain alliance are to deny the Tories a majority, then they will need to win seats such as South East Cambridgeshire, which is the Unite to Remain target 249 and Conservative target 310 – probably enough to put Liberal Democrat leader Jo Swinson in 10 Downing Street with support from the SNP and the remains of the parliamentary Labour Party.

Winning in constituencies such as this shouldn’t be impossible – the Lib Dems came within less than 6,000 votes of victory in 2010 and the Greens kept their deposit by winning 5% of the vote in 2015. But it does show how difficult denying the Conservatives a majority will be through a Lib Dem/Green pact rather than the traditional route of Labour winning in Labour/Tory battleground constituencies.

Even with the Brexit Party polling in the mid-teens, the Conservative Party have the upper hand over their electoral rivals in the current polling environment. Unless the Labour Party make another Lazarus-style recovery like they did in 2017, stopping the Conservatives from regaining their majority with a Remain alliance would require an unprecedented revolution at the ballot box.

The data used in this analysis can be found here.