Pons and Batle's enumeration conjecture for tree-child networks

Let $\mathcal{TC}_{n,k}$ be the set of all tree-child networks with $n$ leaves and $k$ reticulation nodes. The numbers $a_{n-1,k}$ are the coefficients appearing in the paper's preceding enumeration, and $n!/(n-k)!$ is the falling factorial factor. Pons and Batle's conjecture.

$$|\mathcal{TC}_{n,k}|=\frac{n!}{(n-k)!}\cdot a_{n-1,k}.$$

This conjecture concerns the enumeration of tree-child phylogenetic networks by leaves and reticulation nodes. The source presents it as a conjecture of Pons and Batle; its resolution is not established by the supplied material, although the paper's title indicates that it proves a conjecture on young tableaux with walls rather than necessarily this enumerative claim.