Uniqueness in the local Donaldson-Scaduto conjecture

The local Donaldson-Scaduto conjecture predicts the existence and uniqueness of a special Lagrangian pair of pants with three asymptotically cylindrical ends in the Calabi-Yau 3-fold $X\times {\mathbb{R}}^2$, where $X$ is an ALE hyperkähler 4-manifold of $A_2$-type. The existence of this special Lagrangian has previously been proved. In this paper, we prove a uniqueness theorem, showing that no other special Lagrangian pair of pants satisfies this conjecture.