Central limit theorem for a random walk on Galton-Watson trees with random conductances

We show a central limit theorem for random walk on a Galton-Watson tree, when the edges of the tree are assigned randomly uniformly elliptic conductances. When a positive fraction of edges is assigned a small conductance $\epsilon$, we study the behavior of the limiting variance as $\epsilon \to 0$. Provided that the tree formed by larger conductances is supercritical, the variance is nonvanishing as $\epsilon \to 0$, which implies that the slowdown induced by the $\epsilon$-edges is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in $\epsilon$.