Here is a sample total range before the action OTR, a tight one to illustrate the point (with 99 in hole, not JJ):
Board: Q♦5♥8♠ 2♥ Q♥
Equity Win Tie
MP2 23.73% 22.89% 0.85% { 99 }
CO 76.27% 75.42% 0.85% { 88+, ATs+, KJs+, QJs, T9s, AQo }
In this case, he is not going to bet with with 9 combos of Axs, but will probably call a bet. If you bet, he will not be able to bluff with 3 combos of KJs and 3 combos of T9s (if that is what he had in mind). So you might net an extra bet against 3 combos by bet-folding, but be warned, see below.
He'll be raising with 3 nut flushes and 1 QQ. What he does with the 9 combos of trips and weak flushes, I don't know. He should probably raise one combo as a bluff. On balance, against this range, it is too risky to bet-fold. The point is that it is still close, in spite of the fact that he is a 76% favorite.
Now lets say he is looser and a little unbalanced and plays sub-optimal post-flop. By that I mean that he fires a second c-bet when he shouldn't, and doesn't have any small SC (T9s gone).
Board: Q♦5♥8♠ 2♥ Q♥
Equity Win Tie
MP2 40.87% 40.36% 0.51% { 99 }
CO 59.13% 58.62% 0.51% { 88+, A8s+, KJs+, QJs, AJo+, KQo, QJo }
His second c-bet with most of the A high are bad because no better hand is folding, and few worse hands are calling. He isn't a favorite and, besides, he has showdown value. Now there are plenty of Ax that will call a bet on the river, but will not fire a third bullet (he isn't that bad). His range still a favorite against 99! From his point of view, which hands are better to raise with (as a bluff) than to call with? I'd say that none is. There are 30+ combos that you might gain an extra bet against (compared to check-calling) if he never bluff raises on that river when you bet. Remove a few combos that now cannot bluff by betting.
The typical situation in micro stakes is when you have initiative. Then most players automatically will bet-fold things like second pair on the river (as a slight non-favorite). It's no issue. What this thread is about is bet-folding without initiative a k a donk-folding, which is (perhaps rightly so) considered fishy.
I have the feeling that initiative is taken too seriously when all the cards are dealt out. The time for semi-bluffs is gone, the way we got to the river doesn't matter, except for putting opponents on ranges, not only their total range, but split up as in the previous post.