I believe that even if my AH was to stop drinking today and never have a drink again, it would take him very long time to stop lying to me. As I see he's too used to it, he can't function around me any other way, it's his defence mechanism. In my opinion the person who is quitting drinking has so many issues to deal with, kind of learn everything from the beginning and it is unreasonable to think s/he can do it all at once.
I even think it'd be easier for my AH to stop drinking than stop lying to me, as I know deep inside he must know I'm the one he has failed the most, and that must be the hardest one to face, as by doing it he'd have to take full responsibility for everything he did... and in our case I don't think that's likely to happen ever... But if I were you, I'd give him some time, cause the way I see it you have nothing to loose...