Can't run the game offline ("Steam servers too busy")

Tried to start the game up in offline mode today and got a "The Steam Servers are too busy to handle your request ... Error Code(3)" error. Bit weird as in offline mode I wouldn't think there was any need to contact the Steam servers.

Anyway, plugged an ethernet cable into my laptop, went online, started up the game and it ran fine. Then I disconnected the cable, restarted Steam in offline mode and tried again, and again I got the "Steam servers are too busy" error.

I've found a couple of posts about similar issues from 2014, but they seem to have been resolved, so I think this is something new.

Any suggestions?