It seems to me that there is some amount of "having your cake and eating it too" in your question.
First of all, I think private mathematical correspondence, including sending preprints, should only be "friendly" (or "collaborative"); it should not be "unfriendly" (or "competitive"). If someone emails me a paper, I don't in general have an obligation to read it. If you think I am about to prove something very soon and you send me something with the aim of taking away the independent nature of my proof, that seems quite unfriendly. The correspondence doesn't advance the field in any way; it only seeks to advance you at my expense. As others have said, if the natural state of affairs is that both you and I will arrive at proofs of the same result independently in roughly the same time frame -- great, we will both get the credit, and it will not be divided in half in any clear way. I don't see why you need to perform machinations to try to prevent that.
Also, you are trying to establish priority in a rather strange way when there are two much more standard ways: submit to a journal and/or post on the arxiv. Regarding this, you say:
My paper on this is however not ready to be put on the arxiv, because there are natural questions related to this conjecture that need to be solved first (otherwise the paper will look incomplete).
But you also say:
I now managed to prove the conjecture. I think it is an important result.
If the conjecture is important, then a proof of it should make for a very nice paper. Most papers "look incomplete" in the sense that they leave something natural undone. And most of us do not get to work out all the consequences of our results in the same paper as those results are first published. Doing too much in one paper is risky for several reasons, one of which is that the longer you sit on a given result, the larger the chance that someone else will do it. You seem to be trying to circumvent this natural tension with your private correspondence. While not unethical per se, I don't expect it will be well received.
Given the situation and the concerns, I suggest that you do upload to the arxiv a short note that proves your conjecture and that does not have all the bells and whistles you would like to eventually have. You say that it is two pages long, so that I would expect that the relevant mathematical community will be able to quickly ascertain whether it is a correct proof of the conjecture. Because of this you don't have to submit right away or even submit the version of the paper you put on the arxiv. You can take the time to make the additions that you want. Of course, other members of the community can respond to your arxiv preprint and build on it as well. That's what's fair, it seems to me.