I'll say it again, because it answers all of your hypotheticals:
RAW doesn't say one way or the other, so make a ruling for your table.
I'm not convinced that the rules as written don't say. Specifically, the rules say when you can or can't keep concentration, and so far as I can tell, they don't say anything that would even hint at "but you can't maintain it through a short rest".
They don't explicitly say that you can, but you know, the rules never explicitly state that you can move on your turn even if someone in your party is named Greg, either. It's just that there's a rule that you can move on your turn, and while there are exceptions, none of them are "unless someone in your party is named Greg".
There are listed things that break concentration. A "short rest" is not one of them.