NL10z Top set on connected board v reg

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NL10z Top set on connected board v reg

Blinds: $0.05/$0.10 (6 Players) BN: $10.11
SB: $4.13
BB: $11.81
UTG: $14.40 (Hero)
MP: $9.04
CO: $8.70
Preflop ($0.15) Hero is UTG with K K
Hero raises to $0.25, 4 folds, BB calls $0.15
Opponent is a reg, but dont have any further notes
Flop ($0.55) Q K J
BB checks, Hero bets $0.18, BB calls $0.18
Think Flop is pretty standard.
Turn ($0.91) Q K J 7
BB checks, Hero checks
Now since im up against a reg, I thought about this hand making a perfect case for a checkback, since were usually so ahead of him and we block his calling range significantly (if I had QQ, or JJ, id be more inclined to double barrel). I would definitely just continue against a fish.
River ($0.91) Q K J 7 3
BB bets $0.81, Hero calls $0.81
River I was unsure about reraising small for value against his two pairs, but surprisingly found that we have around 42% equity if he bets all of his straights and twopairs here (apparently our KK blocks a lot of his combinations), so thats not really a move and call is the only play.

I plugged this into solver to get an idea and it seems like it would just like to continue betting here, and rather checks back something like AK. Reason being is that we should still expect to get a call from a lot of pair+OESDs and what I think is more important, villain has a lot of two pairs in his range that dont want to play aggressively OTF (prob since we have all the sets and he doesnt) that we get heaps of value from.

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