I'm assuming everything we beats folds always, and everything that beats us calls. I don't think he's folding any flushes, sets/straights re-raise before the river the vast majority of the time.
So I think Check calling > jamming.
Might be a check fold though, depends on how often he's turning made hands into bluffs here once you check. Which good players till be doing a ton, and weak/bad players will be doing almost never.