Your resignation letter is one of the very few things that follows you via PRIA, other than a very high level overview of your training records. And HR people are always screening for “problem children” with the potential for “discipline issues”.
G7 (and AX for that matter) might be/have been considered bottom-tier regionals by a lot of pilots, but the big 6 have hired plenty of pilots from both places as far as I’m aware, so I doubt they think particularly poorly of TSH.
It’s just such a silly thing to get yourself weeded out over at a potential future employer, even if I’m sure it felt great in the moment.