Why are they not always in a screen or tmux session?
It seems that your unwillingness to take that precaution has lead to your threatening physical violence on co-workers. It can also turn momentary network hiccups into a much more serious deal than it would otherwise be. Perhaps you should think carefully about that fact.
Anyone have a suggestion how to use tmux for those?
It's my desk and my Ethernet cable and it's plugged into my laptop. No one else sits at my desk and no one should unplug my Ethernet cable from my laptop when I step out of the office. Period.
Edit: So is an office with three desks (one desk assigned to each worker) an "open-plan" or a "shared working space"? Three people work in that office and they all have a key to the door.
If none of this is true, you might as well be talking about purple dogs stealing the umbrellas from your hamburger patch.