So if hero shoves, the question is how "pair heavy" is villian when he 3 bets the open raise. If he's 3 betting k hi and q hi hands I favor the shove, if he's being a somewhat selective 3-bettor, I favor a fold.
Calling might be okay but it maked playing the hand tougher. The pot odds pre are 9 to win 16, not a great direct price on set mining and hero will have 62 behind in a pot of 25 for an spr of about 2.5:1, which isn't a great implied price either and make for other awkward decisions.
So to call, hero has to play beyond set mining, there has to be flops where hero can win this pot without improving. Hero is going to have to bet boards that miss villian's unpaired hands.
This is tough, 44 is probably not the bottom of our opening range as hero so it feels too strong to fold, but it doesn't feel strong enough to shove, and it seems too awkward to call.
So I am just saying hero has to be really good post flop to call. Or really sure villian can fold to the four bet shove a decent hunk of the time to take that line.
So I guess my answer is it depends. Personally I am probably opening enough preflop where I need to at least defend with 44. If I shove, it is because I think villian will fold a lot of coin flips and maybe 55,66,77 (if those are even 3 betting hands for villian.). I anticipate being called by at least 88+, AK, AQ, maybe AJ, AT, A9, and KQ. So I would be hoping villian has KJ, KT, QJ, QT in his 3 betting range and would fold those.
I think, despite the awkwardness of the stack, I have talked myself into a call here and try to shut villian out on flops that miss his unpaired hands.