Is the US Senate required to hold a trial after the House Impeaches?

The US constitution grants the Senate the power to try an official who is successfully impeached in the House:

But it’s not clear from the wording if the Senate can decline to hold a trial. As a purely hypothetical exercise (ha!), suppose the House were to impeach the president, but the Senate is controlled by the party to which the president belongs, and the leadership decides that it doesn’t want to hold a trial.

What happens? Has this ever happened at the federal level, even if not for the president?

I was going to post this in GQ, but I’m not even sure there is a factual answer to this.

No*
The Senate traditionally does not like to hold trial for someone who resigns. Thrice has someone impeached and the trial was dismissed due to his resignation: Mark Delahay, George English and Samuel Kent. More specifically to the OP, it was an open question for the Senate if they could even hold trial on someone who resigned. The trial of William Belknap was significant no only because it was the only impeachment of a cabinet member but also that the Senate decided they could hold trial for someone who resigned since upon a conviction they could bar the person from ever holding office again.
*The question is specifically answered in the case of William Blount. The House impeached and the Senate refused to accept it,

Well, that’s an interesting twist that I didn’t even think about!

So, let me refine my question: Let’s assume that the president does NOT resign after being impeached.

I’d assume it would go to the Supreme Court to decide if they want to force the Senate to hold a trial.

If you can’t get 51 votes to force the Senate to hold the trial, how are you going to get 67 votes to remove?

The only reason the minority party would try to force a trial is to get people on record.

And even that wouldn’t resolve anything, because the Supreme Court has no enforcement power over the other branches. Recall the decision that Native tribes could not be dispossessed of their land, to which Andrew Jackson allegedly responded to the effect: “The Supreme Court has made their decision; now let them enforce it!” whereupon the Trail of Tears ensued.

I don’t see how SCOTUS or any other body has the slightest power to force the senate to do anything.

Agreed. They might have some sense of “moral authority”, but then so does the Senate. They could simply say: Your opinion is very important to us…

And I was actually thinking of a real case (hence the “ha!”) where the House flips to Democratic, they impeach Trump, and the Republicans in the Senate decide to pull a “Garland” and do nothing.

It would be a nonjusticiable political question. The Senate has the right to determine its own procedures and rules, and if they don’t want to put the matter on the calendar, that’s their business.

I could have sworn Clinton was impeached by the House and it never went any further than that, but Wikipedia says he was acquitted by the Senate, so never mind.

And even if the Supreme court wanted to expand their reach and rule on things that aren’t in their purview, the OP would be an awful case for it so they wouldn’t do it out of practicality. If they did flex their muscles and risk their reputation to tell the senate that they have to hold a trial, the senate that can’t get 51 votes to hold the trial certainly isn’t going to get 67 to convict, so forcing them to hold a ‘he’s innocent’ trial wouldn’t do any good. When the Supreme Court pushes into new or questionable territory, they do it because they think there is a good reason to do so, not just on a whim, and in this case the Senate holding a trial would not actually accomplish anything at all.

I think it was pretty clear Clinton was not going to be forced from office; they would have had to get a lot of Democratic votes to do so but got none, not even Holy Joe Lieberman. The Republicans seems to have calculated the the public embarrassment of the trial would help them with voters; it didn’t work out that way. So in this case the Senate made a conscious choice to go forward with a “he’s innocent” proceeding.

Johnson’s impeachment, OTOH, turned on a single vote.

Only the people have the power to force the Senate to act through the electoral process. That leaves a question of the endurance of a call for impeachment by the House. Would it survive an election that changes the Senate’s mind if they had failed to act initially? Though only a third of the seats in the Senate could change hands in a single election, there could be more than one during a presidents terms in office, and it wouldn’t require a change in the majority of seats, just a few incumbents voted out could change the minds of the others.