LONDON — Support for a second referendum on Brexit is growing among British voters, amid diminishing optimism about the U.K.’s future outside the European Union and waning confidence in London and Brussels’ handling of the divorce negotiations, according to a detailed new poll shared exclusively with POLITICO.

Just over half of those surveyed said they back some form of a second referendum, with the most popular scenario being a vote to either accept the government’s Brexit deal, or to stay in the EU — an option backed by 34 percent. That is up from 28 percent in a similar survey in March, according to new findings from Greenberg Quinlan Rosner Research (GQRR). In the earlier survey, 45 percent of people backed a second referendum — although these figures include public votes that would still mean the U.K. leaves the EU under either outcome.

The pro-EU Liberal Democrats are the only U.K. political party now backing a second referendum, but they hold just 12 seats in parliament. The opposition Labour Party, which meets for its annual conference in Brighton this week, has so far ruled out the idea — and decided Sunday not to have a debate or vote on the issue during the gathering. The Scottish National Party is not formally pushing for a vote, but leader and Scottish First Minister Nicola Sturgeon hinted it still might, telling the New Statesman last week that the case for a second vote "may become very hard to resist.”

The Labour Party and the Department for Exiting the European Union declined to comment on the findings.

GQRR interviewed 1,203 people between September 11 and 13, 2017. Increased interest in a second vote since GQRR’s last Brexit survey in March — the month Theresa May triggered Article 50 to formally start the Brexit process — coincides with mounting pessimism around the Brexit talks, which enter their fourth round in Brussels this week, and about the U.K.’s future outside the bloc.

Among Labour voters who responded to the survey, support for some kind of second referendum is at 70 percent, with a referendum on accepting the Brexit deal or staying in the EU the most popular, backed by 51 percent.

Labour MP Ben Bradshaw, supporter of the pro-EU Open Britain campaign group, said British voters would not "roll over and allow the government" to deliver Brexit "without having a say on the deal."

“People have every right to keep an open mind about Brexit and to judge for themselves if the deal the government brings back matches up to what they have been promised by Leave campaigners and ministers," he said.

Overall, 39 percent of respondents reject the idea of a second referendum — and 7 percent want a "hard Brexiteer’s referendum" in which people would choose between the government’s deal and leaving the EU without a deal. Another 10 percent back a referendum that would give voters the chance to reject the government’s Brexit deal, but send it back to the negotiating table.

The country is still split down the middle in its wider attitudes to Britain’s Brexit future, with 47 percent of those surveyed saying they are “worried” about Britain post Brexit, against 46 percent saying they are "hopeful." But the trend is downward: 41 percent were worried and 50 percent hopeful when GQRR asked the same question in March.

Since then, Prime Minister Theresa May's decision to call a snap election has backfired, as she has lost her parliamentary majority, and the first three rounds of Brexit talks have yielded little progress on key areas of dispute, in particular over the U.K.’s financial obligations to the EU. The poll was carried out prior to May's Florence speech, in which she called for a minimum two-year transition period during which the U.K. would abide by EU rules and pay into the budget in exchange for unfettered access to the single market.

Those surveyed express more frustration with the EU’s approach to the Brexit talks than with May and her Brexit Secretary David Davis. While a quarter of people think neither side is being “fair and reasonable” in the talks, a larger group — 36 percent — think the British are being fair and reasonable and the EU is not. Only 13 percent think the U.K. carries a greater share of the blame for talks not progressing.

On Brexit bill, ECJ and Irish border

Voters also back the government’s challenge to the EU over the Brexit divorce bill. The prospect of the U.K. paying £50 billion — a figure cited by some EU officials — to settle its financial obligations to the EU is rejected outright by 61 percent compared with 23 percent who would accept such a figure. A lower bill, of £30 billion, would also be deeply unpopular, with 54 percent saying it would be unacceptable and 29 percent in favor. Among Remain voters, a £30 billion bill is considered acceptable, the survey suggests, but £50 billion would be rejected even by a majority of them.

There is also support for the government’s refusal to accept the direct jurisdiction of the European Court of Justice (ECJ) as part of any Brexit deal. Some 59 percent of respondents said the U.K. should not be bound by the judgments of European courts after Brexit, and even among Remain voters the issue was split, with 43 percent accepting that European judges should have a say in British law after Brexit, and 42 percent opposed.

On the issue of the Northern Ireland border, only 31 percent said customs checks between the Republic of Ireland and the North would be acceptable, with 47 percent against.

While there is broad support for the fundamentals of the U.K. government’s negotiating position, it is the increasing interest in revisiting the entire question of Brexit that will be of most interest, particularly to the opposition Labour Party, which gained much of its support from voters drawn to its softer stance on Brexit. The Lib Dems will also take heart from the findings, after failing to cut through with voters at June’s general election with their second referendum message.

Responding to the poll, the party’s Brexit spokesman Tom Brake said support in the country for a vote on the final deal was clearly growing. “If the Brexiteers are as confident as they make out, why are they running scared of the facts and of the British people?” he said. “Perhaps it is because instead of the land of milk and honey promised by the Leave campaign, we are already seeing falling living standards and jobs moving abroad.”

But Professor Anand Menon, director of U.K. in a Changing Europe said, these are "depressing findings overall for remaining Remainers and those in favor of a soft Brexit. Public opinion seems hostile to things that would keep us in the single market, to voting again and to paying what it seems the EU might ask us for. Strikingly, all this despite confidence in the outcome waning."

The source for all graphics is GQRR's September 2017 poll. Percentages may not add up to 100 percent due to rounding.