AK in a 3-bet pot. Should I value bet river?

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AK in a 3-bet pot. Should I value bet river?

Blinds: $0.25/$0.50 (4 Players) SB: $67.55
BB: $52.86 (Hero)
CO: $10.00
BN: $8.25
SB is TAG, good reg.
Preflop ($0.75) Hero is BB with K A
CO folds, BN calls $0.50, SB raises to $2.00, Hero raises to $6.00, BN folds, SB calls $4.00
Flop ($12.50) 2 2 K
SB checks, Hero bets $4.13, SB calls $4.13
Turn ($20.76) 2 2 K J
SB checks, Hero bets $12.50, SB calls $12.50
I should bet turn, right? Or does checking make sense?
River ($45.76) 2 2 K J 9
SB checks, Hero checks

I was setting up the stack sizes to shove river, but at the end decided to not value bet on this particular river. I can't see a lot of worse hands in Villain's range which will call me. Maybe KQ, but that's it.

Meanwhile, Villain could have KJ, QTs, JJ and occasionally A2s (though A2s will likely raise on an earlier street). I don't block sets or straights.

Should I value bet river?

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