[Nick C]

Compared to calling down, we're charging ourselves to draw versus 66. (It's possible Villain won't fire with that again on the river, opting to check-call instead -- or maybe even check-fold, if he gets counterfeited.) And if Villain will fire the whole way with A7s, we'd prefer to let him keep doing that (though it wouldn't be a tragedy if he folded his kicker and chopping outs, especially since he might not really fire again). Also, there is some chance he's really got a jack and persistently going for a bet/3-bet line.

However, as compensation, I think you've got some good fold equity versus the marginal part of Villain's range. And, if behind, you could have as many as 16 outs.

I have trouble figuring out when to make free-showdown raises. I think I like this one, though.