Meissner-polyhedron minimization among constant-width bodies containing an extremal set

Prove that for every extremal finite set X of unit diameter, the minimum surface area among all convex bodies of constant width one containing X is attained by a Meissner polyhedron based on X.

Background

The paper notes that Meissner polyhedra are dense among bodies of constant width, but density alone does not establish that a surface-area minimizer subject to containing a prescribed extremal finite set belongs to the Meissner family.

This conjecture asks whether the constrained minimum over all constant-width-one convex bodies containing X can always be realized by one of the finitely generated Meissner polyhedra associated with X. The equivalent volume formulation follows from Blaschke’s relation.