This problem is worked around by using the seldom used HP-provided software "HP scan".
This software needs to be installed manually on every single PC this problem occurs.
Since it is bunched in with other software that doesn't see much use and it is a hassle to use (e.g. it doesn't always automatically choose to save the whole document, but one needs to choose the area of the scan that is saved manually) I wager it's an oft overlooked program.