Polytopality conjecture for pure intervals of the $s$-weak order

Let $s$ be a weak composition. For a pure interval $[T,T+A]$ of the $s$-weak order, consider the poset of pure intervals contained in it and the restriction of the $s$-weak order to it. Polytopality of pure intervals. The pure intervals are polytopal in the following sense: there is a polytope $P$ of dimension $|A|$ such that

1. the inclusion poset of pure intervals contained in $[T,T+A]$ is the face lattice of $P$; and
2. the Hasse diagram of the restricted $s$-weak order is the edge graph of $P$.

This conjecture would show that every cell of the combinatorial complex formed by pure intervals admits a convex polytopal realization, a necessary step toward realizing the entire $s$-permutahedron as a polytopal complex.