The problem is, the questions surrounding potential testimony remain unanswered. If there was a problem requiring judicial resolution in the House, wouldn’t there also be a problem when the Senate seeks testimony?
The answer is yes, but possibly in different ways. First, it seems extremely unlikely that a court would rule that the president’s aides have a blanket immunity that would prevent them from even appearing before the Senate. But, at the same time, it seems likely that some, or perhaps all, of the conversations between Bolton and the president, or Mulvaney and the president, would be covered by executive privilege.
There are cases in the past, most notably a matter involving George W. Bush White House counsel Harriet Miers and chief of staff Josh Bolten, that covered similar issues. That case ended in a way that was inconclusive for Congress: The House held Miers and Bolten in contempt, and a judge ruled that they had to appear for testimony. But the ruling did nothing to prevent them from appearing and refusing to answer questions based on executive privilege. And then the case dragged on past the end of Congress’s session, and the once-passionate issue petered out.
So, that is not exactly precise guidance for the current situation.