I completely agree with the first paragraph of fedja's answer: the only crucial point is that you need to be clear that these assertions are not needed for the proofs of any of the main / formally stated results of the paper.
In order to figure out what to do with these "probably true but not fully checked" assertions, I suggest that you think about your purpose for including them. From your comments, it sounds like these are stronger forms of intermediate results, where "stronger" means stronger than you need to prove your main results. What to do here is an interesting problem that may (or may not; I am curious) be relatively specific to the academic field of mathematics. One aspect of this to address is:
Why not include the stronger forms of these results?
In my opinion, all other things being equal it is a good thing rather than a bad thing if one's lemmas are stronger than strictly needed for the theorems they are used to prove. Virtually all academic work is done with the hope that it will be useful to other academics in the field, and this hope should include methods rather than just results. If the idea of proof of Lemma X also proves more general Lemma Y, it may well be more efficient in the long run to state Lemma Y from the outset, as otherwise there may be future mathematicians having to state and prove Lemmas Y.1, Y.2 and so forth. (Now these mathematicians may simply say "The proof of Lemma Y.n is similar to that of Lemma X and is omitted," but I don't really like this and I am not the only one.) This is especially true if you plan to make use of Lemma Y later on. For you it sounds like that is not the case.
However it is not always a good idea to follow the above procedure. If you have 20 pages of lemmas to prove a theorem when in fact 5 pages of lemmas would prove that theorem, then a different kind of disservice is being done to the reader (and one that referees are especially likely to object to). Thus I think that realistically you have to evaluate the risk/reward in every case. However I will make the following suggestion: If at all possible, you should write out a precise statement and proof and then decide whether to include it. I hope it doesn't sound paternalistic when I say that this is the kind of thing that sounds very onerous until you get into the habit of doing it, and then it is often not so bad. Another perspective on this is: you are never going to have a quicker and easier time writing down the proof of Lemma Y than when you have just written the proof of Lemma X.
This is a SE answer rather than an essay, so I will have to stop soon, but I think there are yet other interesting aspects of this question. Here is a hint of one: how valuable it is to "write down the most general version of your lemma" seems surprisingly field-dependent. I am a number theorist who is very algebraically minded (I am happy to make active use of commutative algebra), but often my work gets combined with other work in a more analytic style. It seems to me that in algebra there often is an "optimal result that the idea of proof of Lemma X actually proves" whereas in analysis this may really not be the case. There are other branches of mathematics too...