So, I just dug through the event log, and every time I switch users when the touchscreen is not working, I see things like this :
A pointer device does not have a product string.
A multi-touch device reported inconsistent contact information.
A touch device does not support the maximum count property.
A pointer device did not report a required angular physical range.
Some of them are reported multiple times. This is actually leading me to believe there _may_ be a hardware issue. I have on-site support, and they're sending someone out to replace the screen sometime next week. Will report back on how it goes.
