Recently I'm terribly busy, so couldn't investigate it further, but the problem definitely is not gone, now it's random, sometimes status gets updated correctly, most of time it's not. I've installed the latest development build, removed Aquamail and all it's data and reinstalled again, the problem still remains. For the time being I will go back to the standard Gmail app, I'm too busy to constantly check my email whenever I see phone LED blinking (and it's still blinking because there was a message in the morning, that I read on my PC long time ago).
I've sent debug log to Kostya, hopefully he can find something in it.