That is a really good question, I have been following this thread hoping for a reply also. I would assume so, but you know what an assumption can do. Hopefully we can get an answer either from a Ring moderator or from a customer that has had this happen.
Hey @JamesH and @Brad! This is indeed an awesome question. In the event that the Alarm backup battery is drained from use and the Base Station is then reapplied to power after going offline completely, the system is intended to power up in the last known mode or status.