I imagine the courts would decline to intervene in impeachment on the grounds that it is a political question. Impeachment, conviction and removal from office must necessarily be expeditious and final. You can’t have the matter wind through the courts while the office remains vacant or the replacement can’t act because at some later date the courts will reverse the conviction.
Impeachment proceedings are clear and is a tool particular to the Legislative branch. Anybody telling us otherwise is working on some sort of agenda.
The term high crimes and misdemeanors is not subject to Judicial review, otherwise the judicial branch could rule that fellow judges cannot be impeached.
In this case, the House is the prosecutor and the Senate is the Jury and the Chief Justice is the procedural judge.
“The House of Representatives has the sole power of impeaching, while the United States Senate has the sole power to try all impeachments. The removal of impeached officials is automatic upon conviction in the Senate. In Nixon v. United States (1993), the Supreme Court determined that the federal judiciary cannot review such proceedings.”
— From wiki.