Well Kostya,
You're not going to believe this (because I certainly don't), but I had occasion to access my company Exchange account from one of my personal computers at home, using the Outlook Web App in a web browser, which is a method that I hardly ever need to use.
Imagine my surprise when I looked at the folders and saw a "1" in parentheses next to the Inbox! And this is after leaving work today with NO apparent unread messages (using the full-blown Outlook client from the MS Office Suite). So in one place, nothing unread; in another, one unread. I had just yesterday selected to mark all messages unread in my Inbox, which made me believe it was so (and there was nothing unread visible).
In the OWA here at home, as soon as I selected to mark all read, I did a Sync with AquaMail and it cleared the counter right up!
I don't know what's up with that, but I'm so sorry for making you and probably others scratch their heads regarding a problem that really didn't exist (at least in AquaMail). Thanks to Kostya (great product), mikeone, and Paris Geek for the helpful posts! I'll be reading the forum regularly.