The Frobenius uniqueness conjecture for degree-9 Wahl chains

A Wahl chain is the minimal resolution chain of a Wahl singularity, and its index is the corresponding positive integer $n$. Let

$$\mathbb{W}_9=\{\text{del Pezzo Wahl chains of degree }9\}.$$

Frobenius uniqueness conjecture. A Wahl chain $\texttt{W}$ in $\mathbb{W}_9$ is determined by its index.