A conjecture proved in drafts I am preparing my math research monograph for publication.

In the current draft there is certain conjecture.

This conjecture was proved in my drafts (available on the Web) of the second volume for this monograph.

Should I:


just remove the conjecture from the book because it is proved?
leave the statement in the book titled as a conjecture because it's proof is not yet officially published (and also may be not enough checked for errors)?
write in the book that I am going to publish the proof (either in a separate article or the second volume of the book)?
One of the most important principles of academic writing is that you shouldn't write anything misleading.  If you present the result as a conjecture with no indication that you have already proved it, then you are giving readers a misleading impression (namely that you don't know how to prove it), so you shouldn't do that.


  Should I just remove the conjecture from the book because it is proved?


That's an acceptable option.


  Should I leave the statement in the book titled as a conjecture because it's proof is not yet officially published (and also may be not enough checked for errors)?


Don't just leave it there without comment.  If you are worried that the proof might be wrong, you can address this by stating things a little more tentatively.  As for whether it is officially published, most of your work seems not to be officially published, so why is that an issue for this conjecture?


  Should I write in the book that I am going to publish the proof (either in a separate article or the second volume of the book)?


Sure, why not?  It's probably preferable to deleting the conjecture: if it was worth mentioning as a conjecture, then presumably it's still worth mentioning as a theorem.