In the end it turns out the solution was exactly what all the forums said, you need to plug the PC in to the HDMI port labeled DVI and then change the name of that port to PC/DVI. The secret sauce, of course, was to figure out how the hell to do that. Select HDMI1 in the sources list and hit the info button and then you get a menu which lets you give sources names. Yay for inscrutable, undocumented, procedures which the support staff don't know anything about. (Even knowing I probably had to rename the source they couldn't tell me how to do that).