$10 NLHE 6-max: AA vs Flop Shove Deep Stacked on Wet Board

Villian Stats (VPIP/PFR/AF): 29/23/3

Villian is playing 29/23/3 with a 3b of 8 and 4b of 0.
Have seen him 3b TT & KTs

888 Poker - $0.10 NL (6 max) - Holdem - 5 players
Hand converted by PokerTracker 4

SB: 99 BB (VPIP: 22.54, PFR: 16.90, 3Bet Preflop: 7.69, hands: 75)
Hero (BB): 162.1 BB
UTG: 59.3 BB (VPIP: 60.00, PFR: 13.33, 3Bet Preflop: 12.50, Hands: 18)
CO: 40 BB (VPIP: 75.00, PFR: 25.00, 3Bet Preflop: 0.00, Hands: 8)
BTN: 191.1 BB (VPIP: 29.41, PFR: 22.61, 3Bet Preflop: 8.02, Hands: 561)

SB posts SB 0.5 BB, Hero posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has A:diamond: A:club:

fold, CO raises to 3 BB, BTN raises to 9 BB, fold, Hero raises to 27 BB, fold, BTN calls 18 BB

Flop: (57.5 BB, 2 players) K:heart: 8:heart: 2:heart:
Hero bets 28.7 BB, BTN raises to 164.1

Hero?

Super gross spot. I expect to see AhKx, KK here so I probably am calling? Dont see what else flats a cold 4bet that jams here. This is probably one of those spots that calling or folding is super close. Would have to run a calc for a more detailed answer.

such a gross spot. Shouldn't we fold considering how deep we are??

Stack to pot ratio is like 2.5:1. We're no longer deep. Ship it in and fist pump.