Face-lattice conjecture for partial permutohedra

Let $\mathcal{P}(m,n)$ be the partial permutohedron and let $\mathcal{B}_m$ be the Boolean lattice of subsets of $[m]$. For a chain in $\mathcal{B}_m$, the difference between its largest and smallest nonempty subsets means the difference of their ranks. Face-lattice conjecture. Faces of $\mathcal{P}(m,n)$ are in bijection with chains in $\mathcal{B}_m$ whose difference between largest and smallest nonempty subsets is at most $n-1$. A face of $\mathcal{P}(m,n)$ has dimension $k$ if and only if the corresponding chain has $k$ missing ranks. This conjecture extends the established characterization for $\mathcal{P}(m,m)$ and proposes a corresponding description when $m\ne n$.