Low-dimensional sandpile toppling-probability exponent

Let $\nu$ be the infinite-volume sandpile measure on $\{0,1,\ldots,2d-1\}^{\mathbb{Z}^d}$, let $\mathrm{Av}_o=\{x\in\mathbb{Z}^d:n(o,x;\cdot)>0\}$ be the avalanche set generated by adding a particle at the origin, and let $|x|$ denote distance from the origin. Toppling-probability exponent conjecture. For $2\le d\le4$, there exists $\eta=\eta(d)\ge0$ such that

$$\nu[x\in\mathrm{Av}_o]=\frac{1}{|x|^{d-2+\eta+o(1)}}\qquad\text{as }|x|\to\infty.$$

This conjecture extends the known high-dimensional bounds for the probability that $x$ topples and parallels the corresponding percolation exponent relation. Its status in dimensions $2\le d\le4$ is open.