In a superflat world, I spawned a Piglin, then gave it a name tag and named it. I contained the Piglin so I could keep track of him and not lose him. Of course, since he's in the overworld, he turned to a Zombie Pigman. But he still kept his name tag name, good so far.

But then when I came back after a while, I saw he despawned, and there was no way he could get out of the walls that I surrounded him with. Pigman who never were a Piglin of course don't despawn with a name. When testing this in the nether, the Piglin never despawned, indicating it has something to do with the mob conversion.