Suppose I am trying to prove a certain statement $S$ and find that a certain plausible statement $T$ implies $S$. However, I don't think that they are equivalent, and so there are possibly other plausible similar statements which imply $S$, but my fantasy only produced $T$.

I have posed a question on MO asking if $T$ is true (concrete part) and if it is not, does anybody know anything related to $T$ (vague part). I have obtained an answer with counterexample to $T$ and also a comment with a true statement $T'$, which is indeed related to $T$. Unfortunately, $T'$ does not imply $S$ (or so I think). However, one of the commentators suggested a slightly weaker statement $T''$, which still seems to imply $S$, and such that it "stays clear" of the phenomenon in the counterexample. He also gave his view of how a possible proof might look like.

Now I would like to know if $T''$ is correct. I see merit in the proposed strategy of how to prove it, but I don't think I can do it myself.

Should I accept the answer with the counterexample for $T$ and create a new question about $T''$, or should I edit the existing question so that now it specifically asks $T''$?

Or should I just give up and not flood MO with similar questions?

Another option is to add $T''$ to my other related question, which received very little attention.

PS This meta-question is similar to this one, but my intention is not to change my original question completely, but rather to make the vague part more specific.