Sandpile avalanche-radius exponent

Let $\nu$ be the infinite-volume sandpile measure and let $\mathrm{rad}(\mathrm{Av}_o)$ be the radius of the avalanche set generated by adding a particle at the origin. Avalanche-radius exponent conjecture. For every $d\ge2$, there exists $\alpha\ge0$ such that

$$\nu[\mathrm{rad}(\mathrm{Av}_o)>r]=r^{-\alpha+o(1)}\qquad\text{as }r\to\infty.$$

This predicts a power-law tail for avalanche radii. The source further records the heuristic prediction $\alpha=2$ when $d>4$, but the stated all-dimensional exponent existence remains open.