[Closed] [NL2 FR BSS] Scary card on the turn

Grabbed by Holdem Manager
NL Holdem $0.02(BB) Replayer
Hero ($2)
BB ($2.89)
UTG ($1.98)
UTG+1 ($2.05)
UTG+2 ($3.17)
CO ($2.38)
BTN ($0.82)

Dealt to Hero J♦ J♥

fold, fold, UTG+2 calls $0.02, CO raises to $0.10, fold, Hero raises to $0.32, fold, fold, CO calls $0.22

FLOP ($0.68) 4♥ 4♣ 2♦

Hero bets $0.43, CO calls $0.43

TURN ($1.54) 4♥ 4♣ 2♦ Q♣

Hero ???

Tough spot on the turn. No reads on the villain. I think it's very likely that the Q hit the villain's range. Check/fold looks too weak, check/raise or all-in would isolate me against better hands, bet/fold is not a possibility as I'm pot committed already. So that leaves playing check/call. But check/call would almost cost me my full stack if he bets, so maybe I should shove myself in order to get the little fold equity that might exist.

Equity:

CO 43.18% 42.69% 0.50% 44+, AJs+, KQs, AJo+, KQo
SB 56.82% 56.32% 0.50% JdJh

What do you think? Thank you.