Haase's face-subgroup conjecture for permutation polytopes

Let $G\leq S_n$ be a subgroup, and let $P(G)$ be the convex hull of the permutation matrices in $G$. For a subgroup $H\leq G$, write $P(H)\preceq P(G)$ when $P(H)$ is a face of $P(G)$. For a partition $[n]=\bigsqcup I_k$, define

$$\operatorname{stab}(G;(I_k)_k):=\{\sigma\in G:\sigma(I_k)=I_k\text{ for all }k\}.$$

Haase's face-subgroup conjecture. If $H\leq G$ and $P(H)\preceq P(G)$ is a face, then there is a partition

$$[n]=\bigsqcup I_k$$

such that

$$H=\operatorname{stab}(G;(I_k)_k).$$

The conjecture characterizes the subgroups whose permutation polytopes occur as faces of $P(G)$, extending the fact that stabilizers of partitions always yield faces. Its resolution status is not specified in the supplied source material.