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
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.