This scenario is a bit more about the theoretical understanding of which hands to continue cbet with and how relatively tight 3bet vs flat ranges play out than really the actual hand.

HJ opens 2x, Hero on CO 3bets 6x, folded to HJ who calls. Flop is TT9, hero holds AKc. Imagine the ranges are quite tight, with hero 3betting something like 88+, AJs+, KQs, AQo+ and callrange of the HJ 66-QQ, ATs-AQs, KJs+, QJs, AJo+, KQo. We are ~50 BB deep. Do we cbet here or not?

My thoughts on first trying to reach a GTO/balanced strategy: on the one hand we have a pretty big range advantage so that makes it that we can cbet more often. On the other hand, betting AKo is not likely to accomplish much as better hands certainly are not folding and there are few hands we beat that will trivially call. That is why I am leaning towards a check, but don't we then pretty much announce that we have AK/AQ?

Now you could argue that I should not try to get to a balanced strategy as it won't be the most profitable one on low stakes MTTs. You would probably be right, but I think there is still value in understanding the GTO strategies, identify where the opponent will make deviate and then target that misstake.

The lack of any replies make me think that my opening post is bad/unclear, can anyone indicate what is wrong with it?

Just buy a solver and run the spot to try and find aggregate patterns. Your question isn't really very clear either and doesn't include stack sizes.

Villain's range will have more Tx than you so your ranges are likely closer than you think. No one can give advice here not knowing stacks as that would effect your decisions.

Thanks for your reply. I was indeed considering to buy a solver but they are rather expensive and I am not sure if it is worth the money (yet) for me at this stage. Nonetheless, I think there can be merit in knowing the underlying reasons behind a certain solution than purely the solution itself. For example, in this case that could be: checking behind is better because you want to get as cheap as possible to showdown and there are few dangerous cards.

I did put in my OP that we are around 50 BB deep btw, if this is too vague then let's just say we are exactly 50 BB deep.

Don't read into a lack of replies having to do with you being unclear. I thought what you posted was very clear. Take it as this forum isn't really busy and it doesn't seem there are many people even participating here. I think just basing it off what you wrote(not knowing how early or late you are in the tournament and not knowing if the HJ is loose or tight) I think a check is fine as well as a 1/3-1/4 pot size cbet. If you get called you effectively most likely will get 2 more streets to hit an A or a K as the HJ will probably check the turn. Just my 2 cents.

Sorry I missed you're 50 deep, my bad there.

You can sometimes cbet this having backdoor playability but would mostly be checking since villain will have more Tx in their range and you likely just flat ATs or mix it.

Alright, thanks for both your thoughts.
When we check, we are looking to call off 1 street unimproved I assume, but probably fold vs turn and river bet? What part of our checking range calls both turn and river, 88 maybe?

Depends on the size but yeah this is a good hand to have that can surrender later while defending against smaller sizes and also have backdoor playability somewhat.

Given your actual hand and opponent's range I would check the flop. TBH I might do this with most of my 3-betting hands on this flop-- aside from Tx, 99, or a combo draw. Assuming a generic opponent at more casual stakes without any metagame going on. Just far too many hands in opponent's range that fit somehow. You might not even fold out an underpair with a bet.

I havent run this spot but solvers tend to like small cbets with most of your range on paired boards in 3bet pots. This a particularly connected paired board so maybe it checks more than usually here, but I suspect it just puts out a 25-30% cbet with most of its 3betting range.

Actually given those pf ranges Im pretty certain in a small range bet. There are only two combos of ATs possible for villain to have and lots overcard and underpair hands.

There is still a lot of worse hands than AK that will call a flop bet. KQ, KJ, QJ are peeling one for sure and we have pretty good equity vs an underpair and can keep applying pressure on turn and/or river.