Polynomial-form volume conjecture for partial permutohedra

Let ${\mathcal{P}}(m,n)$ be the partial permutohedron, and let $v(m,n)$ denote its normalized volume. For each relevant $n$ and $i$, let $p_{n,i}(m)$ be a polynomial in $m$. Polynomial-form volume conjecture. For arbitrary $m$ and $n$, the normalized volume is

$$v(m,n)=\binom{n+1}{2}^m-m\binom{n}{2}^m-p_{n,1}(m)\binom{n-1}{2}^m-p_{n,2}(m)\binom{n-2}{2}^m-\ldots-p_{n,n-2}(m)\binom{2}{2}^m,$$

where $p_{n,i}(m)$ has degree $2i+1$ and positive leading coefficient. This extrapolates the sculpting formulas established for $n\leq 4$: the first two terms arise from the first two sculpting steps, while the general form and the stated polynomial properties remain conjectural.