The question is, what does he check the turn with that we lose to? I can't think of anything. Does he really check AK with a spade ip? What about the nuts? Would he really give us a free river if we have sets in our range? Certainly a set or AA is betting. I think villain rarely has a ten here, not often, but not never. He often has JJ and AQ w/spade, and sometimes worse pairs might call after taking this line.

By the river, I think we're ahead way more often than behind, so value betting the river makes sense to me. If he raises, fuck him, call, it's a bluff more often than the nuts.