Symmetry-preserving density of Reuleaux polyhedra

Determine whether, for every symmetry group H of a Reuleaux polyhedron, the family of Reuleaux polyhedra whose symmetry group contains H is dense in the family of bodies of constant width having symmetry group H.

Background

Meissner polyhedra are dense among three-dimensional bodies of constant width, but smoothing one edge from each dual-edge pair can destroy symmetries; for example, no Meissner body has tetrahedral symmetry. Reuleaux polyhedra may retain the required symmetries, motivating the question of whether symmetry-preserving Reuleaux approximations are dense in the corresponding symmetric class of constant-width bodies.