Stabilizer characterization conjecture for faces of permutation polytopes

Let $G\leq S_n$ be a permutation group, let $H\leq G$ be a subgroup, and let $P(G)$ and $P(H)$ denote the associated permutation polytopes. For a partition

$$[n]:=\{1,\ldots,n\}=\bigsqcup_i I_i,$$

write

$$\operatorname{stab}(G;(I_i)_i):=\{\sigma\in G:\sigma(I_i)=I_i\text{ for all }i\}\leq G.$$

Stabilizer characterization conjecture. If $P(H)\preceq P(G)$ is a face, then

$$H=\operatorname{stab}(G;(I_i)_i)$$

for some partition $[n]=\bigsqcup_i I_i$.

The paper notes that stabilizers of partitions provide an obvious class of subgroups whose permutation polytopes are faces. The conjecture asks whether every subgroup yielding a face arises this way; its resolution is not given.