You can be sure that Sunak and Sharp have been on the phone because Sharp is a government appointee, and can only be removed by the government. It's an interesting game being played, because Sunak is trying to present the debacle as an issue for the BBC to resolve, and nothing to do with him.
It all depends on whether the shît starts to stick to the Tories, and my guess is it will because it hasn't gone unnoticed that the people at the top of the BBC are Tories themselves, Tory donors, appointed by Tories, and embroiled in Tory financial sleaze.
This one isn't going to go away because football is intrinsic to British culture.