Benjamini–Yadin cylinder growth conjecture for external DLA

Let $(\mathcal{G}^n)_{n\geq 0}$ be a family of $d$-regular graphs with $|\mathcal{G}^n|\to\infty$. For each $n$, form the cylinder graph $\mathcal{G}^n\times\mathbb{N}$, let $(\mathcal{E}_t)$ be external DLA started from the zero layer, and let $T_m$ be the first time the cluster reaches level $m$. Benjamini–Yadin's cylinder growth conjecture. There exist $0<\gamma<1$ and $n_0$ such that, for every $n>n_0$ and every $m$,

$$\mathbb{E}[T_m]\leq m|\mathcal{G}^n|^\gamma.$$

The theorem preceding the conjecture gives a weaker bound for rapidly mixing regular bases; the conjecture seeks sublinear-than-the-base-size growth per level in general for such graph families.