Wedge conjecture for faces of permutation polytopes

Let $F$ be a face of a permutation polytope and let $F'$ be a face of $F$. The wedge over $F'$ is the polyhedral construction described in the paper, and the complement of $F'$ in $F$ is considered as a face condition.

Wedge conjecture. The wedge over $F'$ can be combinatorially realized as a face of a permutation polytope if, or only if, the complement of $F'$ in $F$ is a face of $F$.

The conjecture is based on explicit checks in low dimensions and proposes a necessary-and-sufficient criterion for wedge realizability. Its general validity remains open.