Checking behind an ace high flush on the river.

[Phoenix1010]

.5/1 NL 9 max on Absolute. Villain appears to be very solid. From memory so a little fuzzy. Do you bet this river?

Hero is MP1 with Ac6c
UTG limps, Hero limps, one LP limp, SB completes and BB checks.
Flop: Tc,5c,4s
SB checks, BB bets $5, UTG folds, Hero calls, others fold.
Turn: Kc
BB checks, Hero bets $11, BB calls.
River: 4c
BB checks, Hero?