On my problem rig I moved my dropping GPU to a 16x PCIe slot on the MOBO as suggested by dlehenky. This seems to have resolved the issues on that rig with stability (so was a separate issue from ssh).
I then decided to do the same thing on one of my stable rigs (one that works as long as I don’t ssh into it) to see what would happen. Unfortunately the same issue is present if I keep an ssh connection to the rig, a GPU will drop. However, it is interesting that it is now a different GPU that drops, and no longer the one plugged directly into the 16x slot.
I don’t run my miner as root and I use Byobu instead of screen. I cant think of anything else that might be different on your setup.
So at this point I have my 7 GPU rigs up and running smoothly, we will see how long they can keep that up.
Hope this helps