I have gotten a little farther in troubleshooting the autotuner in the
FT-1000. It seems that it works perfectly on all bands except 20 meters,
where it won't work at all.

I have no service information for the FT-1000 in general or the autotuner
in particular. Anyone have any idea whether there is some specific thing
that's unique to 20 meters that might have gone bad?

I would have suspected one of the tuned circuits (open L, shorted C) but
the thing doesn't even try to find a match, just sits there showing a very
high SWR and not making any moves at all.

I suppose it could still be one of the LCs, and the thing just shows such
a high SWR that it gives up and doesn't even try (but, it also doesn't
light the "HI SWR" LED...).