That's not the point.
The President has fatally made several assertions which have no basis in law, are not supported by policy, and contradict the Supreme Law of the land. The information he has freely provided today may be introduced as evidence during a war crimes tribunal.
The President cannot claim "privilege" on conversations between White House counsel and DoJ Staff. These conversations supposedly do not involve the President. However, if the President insists on asserting privilege with the intention of Protecting his conversations about the US Attorney firings then he's fatally asserting protection for something that does not need to be proven: Not what he said, but whether he was or was not involved in the decision.
Asserting privilege on the US Attorney firings fatally raises the legal problem for the President: He can only assert privilege for something he was personally involved; and it does not protect the open conversations which have been documented and openly reported involving his White House counsel and DOJ Staff.
White House counsel can be forced to testify under oath. If they refuse to testify, then that refusal can be adversely and correctly interpreted to mean that they are protecting the President for something he supposedly was not involved. The President in asserting privilege has fatally raised the legal issue: Not whether, but how much the President was involved with the decision to fire the US attorneys; whether the President did or did not participate in illegal activity, planning, and other efforts to illegally retaliate against the US Attorneys. The President cannot claim that he has the authority to fire someone; but claim that firing did not involve the President.
The President does not dispute he has the power to replace. The question is whether the president wants to admit that he fired the attorneys illegally for their effort to prosecute the President's political allies who crated shell companies to funnel money to the GOP for partisan purposes. The facts and evidence are not on the side of the President; neither is Congress.
Executive privilege may not be delegated to non-executives; or to shield conversations that the President supposedly said did not involve him, but are supposedly lawful. If the conversations is lawful, then the Staff coordination on this issue – which has been disclosed – is subject to Congressional review.