Sandpile avalanche-size critical exponents

Let $\nu$ be the infinite-volume sandpile measure. For an avalanche started by adding a particle at the origin, let $S_o=\sum_{x\in\mathbb{Z}^d}n(o,x;\cdot)$ be the total number of topplings with multiplicity, and let $|\mathrm{Av}_o|$ be the number of distinct toppled sites. Avalanche-size exponent conjecture. For every $d\ge2$, there exist exponents $\tau=\tau(d),\tau'=\tau'(d)\ge0$ such that

$$\nu[S_o\ge k]=k^{1-\tau+o(1)},\qquad \nu[|\mathrm{Av}_o|\ge k]=k^{1-\tau'+o(1)}$$

as $k\to\infty$.

The conjecture describes power-law tails for avalanche size and avalanche area. The source notes heuristic evidence, including the prediction $\tau=\tau'=3/2$ for $d>4$, while the general assertion remains open.