Paul,
?
I saw very similar symptoms with my mount 3 times over last month. 20 nights, give or take.
?
I do imaging with KStars/Ekos. The mount is connected to PC via Ethernet, HC is not connected at all. At some random point the mount stops talking to INDI driver. I do not see anything interesting the exact moment when it happens, seems pretty random. Meanwhile, the mount continues tracking until it hits mechanical resistance. Gemini Web server does not respond. Ethernet link is up as seen from the PC but no traffic there. When I see it happened, I come to the mount and plug HC. It lights up, goes to 'Connecting to Gemini' screen, waits there several seconds, and cycles back, again and again. Simple power cycling Gemini restores it back to a good state.
?
This has never happened during the two years I have and very actively use the mount. It was L5 originally. I upgraded it to L6 as soon as the upgrade kit became available. I upgraded it further to 6.10 later with GFU. So far so good.
Until recently when bought second mount. Now please bear with me.
?
That mount's Gemini bricked on second night. I contacted Losmandy and they sent me a replacement SD card. You may have heard this story from them. That new card helped resurrecting the Gemini and also likely updated the HC. Likely (I cannot say for sure but also do not see any alternative explanation) I then plugged that HC to the first mount for some quick task as I occasionally do. Nothing wrong at that point but next day the first mount bricked too! Losmandy sent me another SD card and it helped again. It happens to be 6.10 which should be the same version and I already had on that Gemini. The mount worked mostly fine since then. One suspicious thing started after that upgrade:

The power supply is 14.1V, not anywhere near 24V. I did not change it. The second mount sits on the same DC output and is connected in very similar manner. It reports 14.1-14.2V. Anyway, the mount has been working, so I ignored this quirk. Before the bricking and upgrade the first mount reported the voltage correctly.
Not long before I was forced to upgrade I saw the coin battery at ~2.8V. So I decided to change it too while I was there. I do not remember when exactly I changed it though. At first it was reported above 3V, but after several days it dropped to 2.9V, then 2.8V. That may be just a crappy battery or a symptom of something in Gemini, I don't know. Somewhere at that time I had the disconnect issue happened to me the first time.
I ignored the problem and continued using the mount for another several nights. Three days ago it happened again and had been waiting for me until last night.
As the coin battery was looking quite bad by then I decided to replace it again. I checked the old battery by shorting its outputs for 10s and measuring voltage after that. It was in the 2.95-3.0V range. The battery itself and Gemini contacts look clean, no rust or anything. Anyway, I put the new battery in, reset the settings, set time, loaded SRAM back.
I then immediately upgraded to 6.12 with GFU, cycled the power a few times. All looked good, so I started the imaging session.
In two hours it disconnected again! I simply restarted it because I have no more ideas what more I could jiggle. It worked fine through the rest of the night and is still online. The above screenshot it taken just now.
?
I hope you fine some clue it this pile of details!