Short answer this that it's a question that has never been answered. only happened once when one of Grant's cabinet members resigned before the Senate part of his trial started, but the Senate didnt convict so no court ever decided if it was constitutional
Yes this is new ground in some ways. The impeachment house vote did set it in motion. Just because the punishment (removal from office) is off the table, does not preclude having the senate trial. And the Constitution does not address this in between situation.
