€2 NLHE MTT: River nut straight, so fold?

[JohnBoyWWFC]

2nd hand of the tournament. Only read is he called PF before and folded flop. Standard c-bet and barrel when we pick up outs on the turn, then we river the nut straight but do we have to fold to his RR? I think it seems obvious but do we have the odds to call ever?

pokerstars No-Limit Hold'em, 2.25 Tournament, 10/20 Blinds (9 handed) - Converted at http://www.handhistoryconverter.com

MP3 (t2900)
CO (t2940)
Button (t2960)
SB (t3120)
BB (t3000)
UTG (t3000)
Hero (UTG+1) (t3000)
MP1 (t3080)
MP2 (t3000)

Hero's M: 100.00

Preflop: Hero is UTG+1 with K

, Q

1 fold, Hero bets t60, 3 folds, CO calls t60, 3 folds

Flop: (t150) 6

, 4

, J

(2 players)
Hero bets t60, CO calls t60

Turn: (t270) 10

(2 players)
Hero bets t135, CO calls t135

River: (t540) 9

(2 players)
Hero bets t270, CO raises to t620, Hero ???

 

[The Messiah]

Your obv a lot more in the know when it comes to mtts, but what's with your bet sizing, especially the flop, I know it's a dry board or whatever but seems awfully weak and expect you get a lot of agg, thus making life harder on yourself, you obv get some lol calls too but still much prefer bigger sizing.
Folding the straight v an unknown while we're getting 4/1 is prob bad, so we prob just have to sigh call, villain might be thin value raising too consid I'd never put you on a flush, or might feel a raise will take you off a weak holding with air.

 

[duggs]

bet bigger on flop, probably dont need to double barrel this turn tbh, river is a good b/f but go bigger aswell

 

[TEG2300]

I concur on the analysis of bet sizing. On this river a flush is not the only hand that thinks it has you beat IMO the somewhat weak appearing bets have J10 as well as some top pairs raising for value against what they believe to be a weaker hand.
I would like to include that I have roughly no idea what I am talking about