Symmetry-realizing diameter configurations

Establish whether every simple non-bipartite quadrangulation map of the projective plane admits an extremal point set in [?]R^3 whose diameter quadrangulation is the given map and on which the map automorphism group acts by isometries.

Background

Mani's theorem shows that every simple 3-connected planar graph has a convex-polytope realization in which every graph automorphism extends to an isometry. The paper asks whether an analogous realization theorem holds for diameter graphs and Reuleaux polyhedra. The conjecture is stated for simple non-bipartite quadrangulations of the projective plane and requires the full map automorphism group to act isometrically on an extremal point set.