On Monday, the Daily Express reported that a group of migrants from Calais’ “Jungle” refugee camp had attempted to board vehicles at the entrance to the motorway, leading to clashes between police, drivers and refugees.

The story claims that this attempt was inspired by “some migrants fearing border controls could be beefed up immediately in the event of a Brexit”. The camp is full of migrants hoping to make their way to Britain by smuggling across the sea or through the tunnels under the Channel, and at the moment, French police tightly patrol the border to keep it in place.

So what would really happen to this operation if Britain left the European Union?

At first glance, it’s the opposite of what the Express claimed. The French economy minister, Emmanuel Macron, has threatened that Calais’ camp would “move to Dover”, as the French would refuse to police the section of the British border located by the Calais port and tunnels. David Cameron has echoed these concerns.

However, this story was slightly misleading: the French currently patrol our border in Calais as part of the 2003 Le Touquet Treaty, which is not linked to our EU membership. The French could back out of the agreement at any time, but they would need to give two years’ notice. It could happen – but not right away.

Meanwhile, when I visited the camp on Wednesday, aid workers and refugees alike did not see Monday’s events as remarkable, as migrants try to board lorries on a near-daily basis. One asks: “When is the vote? Friday?” Alexandra Simmons, an aid worker with Care4Calais, tells me that “there was a combination of foggy weather and the traffic jam – I could tell a few would try that day”. She adds that during Monday’s clashes police threw tear gas canisters into the camp itself, burning holes in tarps and tents. The camp’s legal centre also says that several refugees were injured, and is currently collecting information on this.

Both the idea of migrants rising up against Brexit, and the idea that Calais’ camp would move to Dover, are scare stories from opposing camps, timed to win political capital. Claire Moseley, founder of Care4Calais, tells me that involving the camp in the referendum the debate is a distraction: “The EU debate should be about people moving within the EU”. These migrants are from outside the EU, and most don’t want to be identified in France at all, because of laws (which Moseley calls “outdated”, and Germany is ignoring) which state you can only claim asylum in the first EU country you arrive in.

However, the negative press around the camp over the past weeks from right wing papers, and avoidance of the issue of migration by the Remain campaign, has contributed to a slowing of aid to the camp. It hasn't helped that in March, the southern half the camp was demolished, and Moseley tells me that people now ask in response to appeals: “Hasn’t the Calais camp gone?”

Alexandra Simmons agrees that the referendum debate has had a “chilling effect” on help reaching the camp, where thousands of men, women and children are still living without sanitation and with limited supplies of food, and around 1,000 new arrivals have arrived in the past four weeks.

The likely outcome, everyone I speak to agrees, is that the referendum will have little effect either way. The camp will still be here, and unless French authorities relent, still will not be recognised as an official refugee camp – though there are rumours one could soon be built in Paris. While this is the case, it is very hard for the refugees to claim asylum at all, and the camp’s poor conditions will continue to take their toll with little outside help.