I heard a constitutional scholar say the if the house had to refer to the courts to remove an objection by the President, that would be asking  the courts  to share if not assume that sole authority.

Why the House doesn't raise that argument every time the President criticizes the House for not taking issues to the courts--and extend the process beyond reasonable time--is beyond me.

If the founders had wanted the House to need the courts in the impeachment process, they could not have used the term: sole authority.