Chan–Robbins–Yuen volume conjecture for the polytope of restricted permutation matrices

Let $n\ge 2$ and let $T_n$ be the set of $n\times n$ permutation matrices $\pi$ satisfying $\pi_{ij}=0$ for $j\ge i+2$. Let $P_n$ be the convex hull of $T_n$, and let $V_n$ denote its relative volume, measured in units of the minimum-volume simplex with vertices in the affine lattice spanned by $T_n$.

Chan–Robbins–Yuen volume conjecture. The relative volume of $P_n$ is

$$V_n=\prod_{i=0}^{n-2}\frac{1}{i+1}\binom{2i}{i},$$

the product of the first $n-1$ Catalan numbers.

The conjecture gives an explicit Catalan-product formula for the volume of this polytope; the paper presents it as the main claim for which evidence is provided, but does not establish the formula.