Catalan product conjecture for Young tableau faces of the Birkhoff polytope

Let $n\ge 2$, and let $F_n$ be the $n\times n$ matrix whose $(i,j)$ entry is $1$ when $j\le i+1$ and $0$ otherwise. The matrix $F_n$ is a union of permutation matrices corresponding to a face of the Birkhoff polytope $B_n$ of dimension $\binom{n}{2}$ with $2^{n-1}$ vertices. Catalan product conjecture. The relative volume of $F_n$ is

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

of the first $n-1$ Catalan numbers. The triangulation computations suggest this simple rule for the relative volumes of these faces, but no proof is supplied.