Sandpile group distribution conjecture for random directed graphs

Let $\Gamma(n)$ be a sequence of $\epsilon$-balanced random digraphs, and let $S(\Gamma(n))$ denote its total sandpile group. Let $G$ be a finite abelian group, and define

$$Q=\prod_{p\text{ prime}}\prod_{k=2}^{\infty}(1-p^{-k}).$$

Sandpile group distribution conjecture. For every finite abelian group $G$,

$$\lim_{n\rightarrow\infty}{\mathbb P}(S(\Gamma(n))=G)=\frac{Q}{|G||\operatorname{Aut}(G)|}.$$

The preceding results establish convergence for every fixed finite set of prime-primary components, but do not control the entire sandpile group; the conjecture asserts that these restrictions are unnecessary. If true, it would also imply that the probability of being coeulerian converges to $Q$.