Ceballos–Pons realization conjecture for the $s$-permutahedron

Let $\operatorname{s}$ be a weak composition. The $\operatorname{s}$-permutahedron $\text{Perm}_{\operatorname{s}}$ is the polyhedral subdivision associated with the $\operatorname{s}$-weak order.

Ceballos–Pons realization conjecture. The $\operatorname{s}$-permutahedron $\text{Perm}_{\operatorname{s}}$ can be realized as a polyhedral subdivision of a polytope that is combinatorially isomorphic to the zonotope

$$\sum_{1\leq i<j\leq n}s_j[\mathbf e_i,\mathbf e_j],$$

where $(\mathbf e_i)_{1\leq i\leq n}$ is the canonical basis of $\mathbb R^n$, and $[\mathbf e_i,\mathbf e_j]$ is the convex hull of $\mathbf e_i$ and $\mathbf e_j$.

This conjecture seeks a geometric realization of the $\operatorname{s}$-weak order, whose lattice structure was proved by Ceballos and Pons. The source does not provide evidence of a resolution.