Normalized-volume conjecture for partial permutohedra with one parameter equal to two

Let $\mathcal{P}(m,n)$ denote the partial permutohedron. Volume conjecture. The normalized volume of $\mathcal{P}(m,2)$ is equal to $3^m-m$. This conjecture gives a closed formula for a family of partial permutohedra whose volumes are otherwise difficult to compute; the paper establishes the analogous formula for $\mathcal{P}(2,n)$ but leaves this family as a conjecture.