Constrained minimization for a regular tetrahedron

Prove that for the vertex set T of a regular tetrahedron of side length one, the minimum surface area among convex bodies of constant width one containing T is attained by a Meissner tetrahedron.

Background

This is presented as a special case of the conjecture concerning constant-width-one bodies that contain an arbitrary extremal finite set. The prescribed set is the four-vertex set of a regular unit tetrahedron.

The problem concerns whether a Meissner tetrahedron is optimal within the larger class of all convex bodies of constant width one containing those four vertices, rather than merely within the finite family of Meissner polyhedra generated by the tetrahedron.