Hard to say without enough data/stats... Some of the players (as you described) tend to lead here with strong draws on the turn, so all the missed flushdraws are definitely in his shoving range on the river (when he 3-barrel bluffs with air, as you said).. Also possible, but rather unlikely, is that he hit is straight on turn or river.. in this case I guess he would play the hand the same way..

I guess when we think the hand through it's a call on the river.. He rather has pure air here and missed draws..Maybe Ah7h, or any other hands with 7h in it.. I would never put him on a J here.. unlikely but also possible could be hands like 89, 87 and 86..

Hard to bluff like this. I would fold. 54 and 85 got there OTR as well.

I hate these spots, but he doesn't seem like a guy who leads turns a lot. Also, 'W$SD when XR/RCB would help'. If I were to guess, it's either a draw, or a hand that improved, but doesn't feel comfortable XR, (J9/J7/A7hh/87hh/89hh etc/maybe some Jx sometimes, but I doubt it) It's a good thing that you block T8 and unblock heart draws.

That said, it's defenitely close otr:

Board: 3h7s9dJh6d
Range 1: J9s,J7s,T8s,54s,J9o,J7o,T8o,98o,Ah7h,KhQh,KhTh,QhTh,9c9h,9d9h,9d9c,9s9h,9s9c,9s9d,9h8h,9h7h,8h7h,8h6h,7c7h,7d7h,7d7c,7s7h,7s7c,7s7d,7h6h,3c3h,3d3h,3d3c,3s3h,3s3c,3s3d
Range 2: TcTd
Equity 1: 71.930% Win 1: 71.930% Tie 1: 0.000%
Equity 2: 28.070% Win 2: 28.070% Tie 2: 0.000%

And I think this hand is probably better to call with than, say, KK or AJ, so I would probably sigh, call and make note.

One read I forgot to mention is that I already saw him donk pot with TP and TOP 2. So I don't think he has a lot of 9x in his range here (except maybe 99).