Nonexistence conjecture for four-dimensional faces of permutation polytopes

Let $Q_1$ and $Q_2$ be the four-dimensional polytopes referenced in the classification table of the paper.

Nonexistence conjecture. There are no four-dimensional faces of permutation polytopes having the combinatorial type of $Q_1$ or $Q_2$.

These polytopes are exceptional among the classified low-dimensional faces because they do not have the facet-complement property shared by the other examples. The paper reports no resolution of this nonexistence claim.