Bet-fold river or just get to showdown?

[BorisTheSpider]

Zero hands on villain.

Is it too weak-tight not betting the river here, or do we risk folding the best hand too often if we bet and he bluffs at it again?

Poker Stars, $0.10/$0.25 No Limit Hold'em Cash, 6 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.

Hero (BTN): $31.35 (125.4 bb)
SB: $25.64 (102.6 bb)
BB: $28.32 (113.3 bb)
UTG: $25 (100 bb)
MP: $27.79 (111.2 bb)
CO: $32.28 (129.1 bb)

Preflop: Hero is BTN with A A
2 folds, CO calls $0.25, Hero raises to $1, SB folds, BB calls $0.75, CO folds

Flop: ($2.35) T 3 6 (2 players)
BB checks, Hero bets $1.75, BB raises to $5, Hero calls $3.25

Turn: ($12.35) 9 (2 players)
BB checks, Hero checks

River: ($12.35) 9 (2 players)
BB checks

Hero?

[griffey24]

Definitely b/f to this action.

[kickass]

I doubt I could fold given action but certainly bet

[Cobra_1878]

I'm betting the turn once he checks, deffo b/f river.