I kind of thought that you need a strong read that the villain is a nit to fold here but I'm now convinced that you almost need a read that villain is capable of some unusual stuff to continue here. I think checking back turn is really bad and results oriented. In the hand I did call. On the river is where it gets weird...
Hero call turn
River (3515): 5
Villain takes about 2 seconds and goes all in for ~1450.
Note that the board paired. If he had a flush on the turn I would expect most live players to always take some time to think about checking and in most cases actually check flushes even ace high flushes. Also, his body language is the opposite of that on the turn. He looks very comfortable, has a sip of his water, and casually helps the dealer count his chips.