It’s hard enough for the leaders of the United States and the European Union to muster public support for the ambitious TTIP transatlantic trade talks.

Now they have the threat of Brexit.

The British will vote in a referendum on exiting the EU in June, raising a huge question mark over talks on the Transatlantic Trade and Investment Partnership.

That is placing ever heavier pressure on the TTIP negotiators, who just opened their 13th round of talks in New York on Monday.

As one of the largest trading economies of the European Union, Britain would play a major role in TTIP, which would create the world’s largest free-trade zone.

TTIP aims to ease non-trade barriers and harmonize bureaucratic rules that impede commerce and investment between the European Union and the United States.

Speaking in Germany on Sunday, US President Barack Obama urged the two sides to push for a final deal by the end of the year, as his eight years in the White House wrap up.

But Britain’s focus has now become its spat with the 28-member EU as the country plunges into a heavily politicized domestic fight over pulling out.

The June 23 vote does not change the goal of the talks, which have been going on for three years.

But for some experts, a British split from the EU could be devastating to TTIP prospects.

“If British people vote to leave the EU, it will put the TTIP talks in shambles,” said Gary Hufbauer, a former US Treasury official now with the Peterson Institute for International Economics in Washington.

“There will be no way of going forward because there will be so many uncertainties.”

Edward Alden, a trade expert at the Council on Foreign Relations, says a British exit would throw the whole TTIP project “into the air” as the European Union and Britain struggle to adjust.

“Conclusion of the TTIP would fall down on the agenda,” he said. “Everybody would be scrambling to try to figure out what is the new relationship between Great Britain and Europe.”

– Stimulus for negotiators? –

Paradoxically, the Brexit threat could push negotiators to accelerate. Former senior US diplomat Daniel Hamilton, director of the Center for Transatlantic Relations at Johns Hopkins University, said that a solid message from the talks that TTIP is moving ahead could remind the British that they will miss out if they exit the EU.

The two sides’ negotiators “will want to go faster to try to influence the public debate in Britain,” he said.

Hufbauer agreed.

“It could be used as an argument to help the ‘remain’ camp to prevail in the Brexit vote,” he said.

Obama, a key driver behind the TTIP pact, accented just that point in speeches in Britain and Germany last week.

If Britain quits the European Union, he warned in London Friday, it would be left behind as EU-US trade relations get a boost from the treaty.

Asked what would happen if Britain did vote to leave, Obama said that although “maybe at some point” it could seal a bilateral trade deal with the United States, “it’s not going to happen any time soon.”

“The UK’s going to be at the back of the queue.”