David,

Suppose one could prove local maxima had configurations that were invariant
under the rotation group of the whole cube. (I am not at all sure it is even
true.)

There are a small number of such symmetric configurations, and they could
probably be easily tabulated. One of them would have to be maximally distant
from home. Thus if we had a QTW solution for each of them, the maximum
over that set would bound God's Algorithm.

I see no reason to believe that a QTW cannot take you between two solutions
that are at the same distance. As DPC pointed out, there are a lot of even
identity paths. E.g., (RUR'U')^6. The two furthest points on the path are (by
symmetry) necessarily equally distant, yet connected by a QTW.

Bill