Free-sum conjecture for centrally symmetric faces of permutation polytopes

Let $F_1$ and $F_2$ be centrally symmetric faces of permutation polytopes, and let their free sum denote the corresponding polyhedral free-sum construction.

Free-sum conjecture. The free sum of centrally symmetric faces of permutation polytopes can be combinatorially realized as a face of a permutation polytope.

The claim is motivated by constructions showing that bipyramids over centrally symmetric faces should again be realizable as faces. The paper presents the broader free-sum realization claim without a resolution.