I confess to not only having no proof to this proposition, but no
longer believing it. The false equivalence of "symmetric" and
"locally maximal" seemed to me visually obvious, given that all symmetric
positions are in fact locally maximal. If one believes this
subconsciously, it is but a short step to the conscious "proof" that
all asymmetric positions are but "on the way" to a symmetric, maximal
one; this of course, is bogus. Now a good definition of symmetric
is also needed here; I assume that what we have been meaning is that
all rotations of the cube are Isomorphic. Note that the Plummer and
Christman crosses don't qualify under this definition. What is a good
definition?