It reconnects by pressing "Connect", of course. All the server apps (web UI, ping, telnet, ftp) start to work by magic when the print is done.
I don't remember, but I'm pretty sure it is a timeout message. The printer's IP is not reachable on any port and protocol. I am and IT guy at the core, so I am 100% about "unreachable". It simply stops being there from the network point of view.
What I didn't check is if the AP still sees it joined at all during the down time. I will do that next time.
I've remembered a maybe important bit: it didn't happen on short prints (<1 hour) and I don't know when it happens on longer prints because I find it dead when I check occasionally.