Facet-removal conjecture for the $s$-permutahedron and $s$-associahedron

Let $s$ be a weak composition with no zero entries except possibly $s(1)$. Removing facets of the $s$-permutahedron. There exists a geometric realization of the $s$-permutahedron such that the $s$-associahedron can be obtained from it by removing certain facets.

This proposes an analogue of the classical fact that the associahedron is obtained from the permutahedron by removing facets. The conjecture is stated only for weak compositions satisfying the nonzero-entry condition, and its general validity remains open.