Monitor has 4 HDMI ports. And even the somewhat lower resolution is still far better than what I had.
Update, if anyone’s curious. Both Windows and the NVIDIA control panel on my home computer are showing the 3840 x 2160 resolution but are showing only a 30 refresh rate. I’ll be investigating this after work today but if anyone has any pointers, I’ll be glad of them.
Meantime, the work computer is showing the 3840 x 2160 with a 60 refresh rate, so all is well there.
I’m holding off on buying a KVM for this. Right now I’m using the old KDVM to switch the keyboard/trackball/camera/headset between computers and the remote control (this bugger has a remote! 
) to switch video inputs. We’ll see how that goes.
ETA: oh, and weirdly, it’s set itself to a scaling factor of 350% (!!) and won’t allow changes. I tried through the advanced settings but it isn’t taking. Windows, I tell ya.