Example scenario @ 6-max 2nl:
We open KJs from UTG and gets flatted by an unknown reg on the BTN. We c/c on K32r flop (we cbet AK/KQ), c/c on 4 turn card, and we face a 3rd barrel on 2 river card.

Do we have to call on the river because we're on top of our range or do we have to make an exploitative fold as an adjustment to the underbluffing reg population?

few things:

preflop raise size matters

flop bet size matters

turn bet size matters

river bet size matters

vs good players it's important to have some strong hands in the check call flop and turn range like 22, 33, 44(vs small flop bet), A5s, A2s at some frequency >0%. If you don't then they can just blast away with big bets for big profit vs your capped range.

If you want to exploit bad players by not having these hands in your check call flop and turn range, then you're basically saying to yourself "this guy is so bad at poker that I don't need to have any strong hands in my river check call range."

Do you have that read? If yes, then fold and exploit. If no, then go back to the flop and turn and see where you could strengthen your river range.

I'm not sure about this one, as I'd often check KQ on the flop as well, and I'm definitely (sigh-)calling river with that. If there was a missed flush draw and I don't block it, I'd be more inclined to call with KJ, but it really depends on sizes and a read on villain. If the player pool is really value-heavy then I suppose you can make a explo-fold on this runout, by my typical thought process for spots like this (when villain shouldn't have many straights, 2pr, trips, or sets that made boats) is "I haz top pair. Top pair is good hand. I no fold top pair."
FWIW KJ shouldn't strictly be the top of your range, anyway. You could be check-calling some combos of A5s and A2s some of the time. Oh, and KK. That's a good hand.

Thanks for the replies guys. It makes sense to me now.

So against a good aggro player, I should protect my checking range by putting some of my strong hands in it so I can call down and so this will discourage him from relentlessly bluffing me when I check because he now knows that my checking range is balanced.

Against a bad player or when I have a strong read on the villain that he underbluffs on a certain spot, then I can just make an explo-fold even when I'm near or on top of my range (only the 1-pair hands).

And when the villain is really an unknown reg, I should base my decision on the board runouts or on the population reg.

Part of the reason you should check some strong pairs on the flop is that your betting range must be able to stand up to serious action. If you include too many draws and thin value hands in your betting range then this becomes a serious problem vs strong players. However, vs players that won’t exploit your mistake it’s probably best to bet early and pick up the pot. Thus vs such an opponent you described above as one that underbluffs, I only have a few hands that can call down all the way. This is exploitable, but I believe it’s best vs mistakes of passivity.

i don't fold kj here

You meant checking range right?

yeah I get it now, thanks.

what's the worst hand you will call here?

pry like QQ with suits diff from K

Yes, pretty much this. But you should also check the flop w/ some strong hands against BAD aggro regs too.
There's a lot of money to be made by inducing bluffs against random button-clickers, especially now that there aren't quite so many drooling calling stations about. i.e. It's pointless betting KK/AK/KQ/KJ/KTs on the flop if villain just folds all his air. Give him a chance to make at least one bluff with his underpair or ace high or JTs-76s, imo, because he's not gonna float with much on that K32r board if you c-bet. Even fish will put you on AK.

you can bet this flop often tho

can you explain why?

No, if you bet too many draws and or value bet too thin vs strong players they will raise you more often and you'll be forced to fold your equity more often. Your turn and river ranges will be weaker after betting the flop and good players can exploit that by betting bigger on the late streets.

However, strengthening the checking range is an auxiliary benefit.

I see. Now I get it.