This is not a criminal trial. Not sure the same rules would apply. IMO, the impeachment will expire with the term of the house that passed it.
I don’t remember the case name but the USSC has ruled, during the House’s hearings on communism in the government and in other cases, that a person’s constitutional rights are not forfeited and that the legislature has to abide by the bill of rights for someone subpoenaed to testify. I don’t see the USSC giving the House and/or the Senate a right to ignore the rights of an accused in a trial, whether judicial or political, when there is a risk that the accused will be punished, either by a contempt charge or removal from office.