In an embarrassing blow to the Cabinet minister, Staffordshire Council said it is up to heads if they reopen on June 1.
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Although the Tory-run council said they hope to get some classes up and running by then.
Councillor Philip White said: “Schools know what is best for their staff and pupils, so it is up to each individual school to make the decision about when and how they can open.
“Our health and safety teams are helping them to make those decisions by giving advice on social distancing measures, and we are also providing every school with a pack of personal protective equipment should someone fall ill.
He added: “Our understanding is that the majority of schools will be open in some way, and will be talking to their parents about what will be happening over the coming days.
“We will continue to support them as much as possible to open in a way that ensures the safety of both pupils and staff.”
The council covers Mr Williamson’s constituency of South Staffordshire.
The Sun revealed yesterday that over 2,200 primaries in England could end up defying Downing Street and stay closed on June 1.
If they all stayed shut, then hundreds of thousands of kids could be delayed in getting back to class.