Shape theorem for single-source stochastic sandpiles

Consider stochastic sandpiles with uniform topplings on $\mathbb{Z}$. Let $\delta_0$ denote the unit mass at the origin, and let $D_n$ be the set of vertices toppled during stabilization of the configuration $n\delta_0$. Single-source shape conjecture. For every $\varepsilon>0$,

$$\mathbb{P}\bigg([-(1-\varepsilon)\frac{n}{2},(1-\varepsilon)\frac{n}{2}]\subseteq D_n\subseteq[-(1+\varepsilon)\frac{n}{2},(1+\varepsilon)\frac{n}{2}]\bigg)\xrightarrow{n\to\infty}1.$$

This predicts that the toppled region, after releasing $n$ particles at the origin, is asymptotically an interval of radius $n/2$; the supplied text gives no resolution.