'eck! My problem just got worse!
Instead of a slightly lighter line, i now have a permanently black horizontal line on the exacty opposite of the screen - at the very top of the icon bar. The pixels stay black even when the iLiad is off, but the brightness of the line changes a bit depending on the surrounding pixels (for example, on a black background, it's a slightly lighter gray instead of black).
Seems, indeed, like a busted controller or wiring.
The good news - the light line on the top disappeared at the same time.
The bad news - the unit was bought from ARINC, not iRex.
Edit: Looks identical to this -
https://www.mobileread.com/forums/showthread.php?t=7358 - except much lower on the screen.