Enumeration conjecture for the generalized sets ${\cal A}_n^j$

Let ${\cal A}_n^j$ be the subset of ${\cal A}_n$ consisting of elements whose first $j$ columns are entirely zero. In particular, ${\cal A}_n^1={\cal A}_n$ and ${\cal A}_n^j\subseteq{00\cal A}_n^{j-1}$ for $j\ge 2$.

Enumeration conjecture for ${\cal A}_n^j$. The number of elements of ${\cal A}_n^j$ is

$$\prod_{i=j}^{n-3}\frac{1}{2i+1}\binom{n+i-1}{2i}.$$

This extends the preceding enumeration conjecture to subsets obtained by requiring initial columns to vanish. The paper gives numerical values and states the formula as a conjecture, without a proof.