Now I've tested to actually use the stuff. Someone sent me a message. I got the message. I replied to the message. I got disconnected. It reconnected automatically. I got the same message again. I replied again. I got disconnected.

The above happened about ten times, but then suddenly it started to work. We talked for a while, and everything worked fine.

But now I have trouble using the ordinary Facebook chat and some other Facebook features in Firefox on my laptop. Going offline on the phone makes it work on the laptop again. I've seen the same behavior using the same plugin in Pidgin on a PC (or it might've been some other IM client with some other Facebook plugin).