Pons–Batle's word-encoding conjecture for tree-child networks

Let $\mathcal{TC}_{n,k}$ be the class of tree-child networks with $n$ leaves and $k$ reticulation nodes, and let $\mathcal{C}_{n-1,k}$ be the class of words over the alphabet $\{a_i\}_{i=1}^{n-1}$ satisfying the stated length, multiplicity, and prefix conditions. Pons–Batle's conjecture. For $n\geq 1$ and $k\geq 0$,

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

This conjectural identity relates the enumeration of tree-child networks to restricted-word encodings and would provide a direct formula for their cardinalities. The source gives no evidence of resolution, so its status remains open.