Theresa May heads to Brussels on Monday for another round of talks aimed at striking a deal about Brexit terms at next week’s EU summit. Mrs May is pursuing a deeply destructive Brexit strategy, which should be opposed, insisting on the UK breaking with both the EU single market and the customs union. However, the recent mood music on money, citizens’ rights and even on the Northern Ireland border has been tentatively favourable to her hopes of a deal. Even so, as the prime minister prepared for her trip this weekend she was faced with a deliberately disruptive new set of demands from her party’s pro-Brexit extremists, who are determined that the European court of justice can play no part in any transitional phase while Britain breaks away from the EU.

Both Mrs May’s diary and the extremists’ demands come right on cue at a significant time. They fully bear out what Alan Milburn said at the weekend as he and his whole team resigned as the government’s advisers on social mobility. Mr Milburn accepts that Mrs May has a personal belief in social justice, and he acknowledges that individual ministers – he singled out the education secretary, Justine Greening – have shown commitment to it in practice. But the former Labour health secretary damningly said it had become obvious that the government as a whole is unable to deliver on the issue.

That is because, he added, there is simply too much focus on Brexit. Because of that, there is not enough space in the government’s brain to allow it to translate nice words about social justice of the kind that Mrs May uttered when she became prime minister into effective actions. Mr Milburn’s resignation letter makes clear that the government has neglected the Social Mobility Commission that David Cameron and Nick Clegg set up in 2012 and, even more importantly, that it is not giving enough priority to the social mobility challenges facing Britain.

Last week, the commission published an immensely disturbing report. It showed that children from deprived backgrounds face growing hurdles in some of the richest parts of the country as well as in the poorest. While London and its suburbs, as well as parts of several cities, are pulling away, some of the most disadvantaged life chances are to be found in wealthy parts of the south of England, including west Berkshire, the Cotswolds, and the Crawley area of Sussex. Of the 65 social mobility “coldspots” identified by the commission, 60 voted to leave the EU in the 2016 referendum.

When that report came out, Mr Milburn said it was essential that mainstream politics offered an answer for those who are trapped by these social divisions. He and his team, who include Conservatives, have now articulated that frustration even more clearly. The problem is only partly that socially concerned Tories don’t turn words into deeds enough. It is that Brexit is not the answer to social injustice that many of the poorest leave voters hoped for. On the contrary. Brexit is a deepening part of the worsening problem.