Revelations about Boris Johnson’s personal life, when they periodically emerge, have tended mainly to serve as tabloid fodder for a few days and then flame out. Two recent allegations, however, could prove much harder for the Prime Minister to simply swat away.

The first emerged over a week ago when The Sunday Times published a story alleging that Johnson, during his time as Mayor of London, used his influence to help American former model and tech entrepreneur, Jennifer Arcuri, now 34, get access to government funds and coveted places on overseas trade missions.

The Sunday Times stories and those that followed, including reports of frequent visits to Arcuri’s flat from the mayor, then married, would have been media catnip at any time. But it’s the allegation that his private life might have impaired Johnson’s judgment as a public official that gives it legs.

They are now the subject of three separate investigations into whether the mayor failed to declare a personal interest, including a police investigation referred by the Greater London Authority because as mayor, Johnson was in charge of policing and crime fighting.

Johnson has denied any impropriety in his official duties. In keeping with his policy, he has refused to comment on the nature of his past relationship with the entrepreneur.

That doesn’t mean the stories will go away. In fact, they could become a major test of whether voters’ concern with Brexit overrides all else in the next General Election.

For many Conservative voters who have been inclined to support his Brexit policy, and especially for many women, the stories will be a reminder of a long-held, small-c conservative view that matters of personal morality and those of probity in public office are often not very far apart.

Arcuri was clearly a force in London’s burgeoning tech scene at the time. Fresh out of business school, she set up a company called Innotech to run events for the tech sector. Johnson’s presence at the events, and hers on mayoral trade missions to Singapore, Malaysia and Tel Aviv, helped put Arcuri on the map in that world.

A 2014 Business Insider story names her as one of the 25 top women in tech, along with Net-a-Porter founder Natalie Massenet and Martha Lane Fox of lastminute.com fame. “Thanks to her close ties with London mayor Boris Johnson, the shaggy-haired politician has repeatedly agreed to speak at the event, which also shows the Government’s interest in the East London tech cluster,” it wrote of Arcuri.

That Johnson would grace those events raises eyebrows now, but it didn’t so much at the time. London’s now established tech sector was young and starving for funds and attention; government officials seemed keen to show their support.

Arcuri is clearly someone who could gather the geeks and the hooded coder-dudes in a room, make them feel they had superpowers, and then go market those powers to politicians and companies that didn’t know their DDoS from their MitM attacks.

She has evangelised for women in tech and tech education. Much of it was also photo-op fodder: Arcuri with Johnson, Arcuri in a selfie with Brexit tsar Michael Gove, Arcuri in front of 10 Downing Street in two separate outfits, Arcuri giving a Ted talk.

The Sunday Times report said that Innotech received two grants in 2013 totalling £11,500 (about $14,000) from a promotional organisation that Johnson was responsible for as mayor. Arcuri reportedly received a £15,000 government grant, under the Sirius programme designed to woo foreign entrepreneurs to build businesses in Britain.

Earlier this year, Hacker House, which Arcuri set up with professional hacker Matthew Hickey to provide cybersecurity training, received the first disbursement in a £100,000 award from the Department of Digital, Culture, Media and Sport, earmarked for British-based companies to provide cybersecurity training and also “boost diversity” in the sector.

The Government has disbursed £47,000 and frozen the rest pending the investigation. Hickey, who tweets as @hackerfantastic and is also Arcuri’s husband, has vigorously defended her and Hacker House against allegations that it won business improperly. Arcuri has said that all funding was in respect to her position as a legitimate businesswoman.

As the Arcuri story was gaining momentum, the first day of the Tory Party conference in Manchester on Sunday was marred by revelations by Sunday Times journalist Charlotte Edwardes, that Johnson had squeezed her thigh, and that of another unnamed woman, at a boozy lunch when he was editor of The Spectator in 1999.

Downing Street issued a statement that “the allegation is untrue,” an unusual move because Johnson doesn’t normally comment on such things. Edwardes tweeted in response: “If the prime minister doesn’t recollect the incident then clearly I have a better memory than he does.”

Some around Johnson at the party conference showed clear discomfort about the whole thing. Health secretary Matt Hancock said of Edwardes, “I know her and I know her to be trustworthy,” a sentiment echoed by former Cabinet minister Amber Rudd. British Chancellor of the Exchequer Sajid Javid came to Johnson’s defence, saying he has total trust in the prime minister.

Together, the two sets of allegations may be harder to shake than any Johnson has faced in the past. Johnson’s ties to Arcuri will have to survive formal investigations, not just trial by media. If Johnson helped Hacker House understand what was required to successfully bid for government contracts, that is one thing; if he failed to declare a personal interest or intervened to see that contracts were awarded, that is entirely another.

The second allegations are more complicated to adjudicate. While Johnson’s personal peccadilloes may be tolerated, Edwardes’s allegations, even if many will question the timing, go further than anything previously reported about the prime minister’s character.

Former defence minister Michael Fallon had to resign over not dissimilar allegations under Theresa May. Her close adviser Damian Green was also forced to resign after being found to have made misleading statements about pornography on one of his parliamentary office computers.

Johnson’s strategy seems to be to tar anyone who discusses such matters as being hellbent on frustrating the UK from leaving the European Union on October 31. Will his divided party toe that line?

His cabinet is reportedly divided over his Brexit plans and leaks on Monday night suggest that the EU is so far not buying his proposals. It may still be that Brexit is so paramount for Conservative voters — or, if not, then at least keeping socialist Labour leader Jeremy Corbyn out of power is — they trump everything else in the next election. Forces are coming together that could test that theory to the limit.

• Therese Raphael writes editorials on European politics and economics for Bloomberg Opinion. She was editorial page editor of The Wall Street Journal Europe