Fulman–Kaplan–Singhal–Warnaar conjecture on ranks of Sylow sandpile groups

Let $G_{\alpha}(n,u)=G(n,\lceil\alpha n\rceil,u)$ be the Erdős–Rényi random bipartite graph with edge probability $u$, let $S_{G_{\alpha}(n,u)}$ be its sandpile group, and let $\operatorname{rank}((S_{G_{\alpha}(n,u)})_p)$ denote the rank of its Sylow $p$-subgroup. For a real number $q$, define

$$(x;q)_i=(1-x)(1-xq)\cdots(1-xq^{i-1}),\qquad (x;q)_\infty=\prod_{j=0}^{\infty}(1-xq^j).$$

Let $p$ be prime, $\frac{1}{p}<\alpha\leq 1$, and let $r$ be a nonnegative integer. Rank distribution conjecture. If $p$ is odd, then

$$\lim_{n\to\infty}\mathbb{P}\big[\operatorname{rank}((S_{G_{\alpha}(n,u)})_p)=r\big]=\frac{1}{(-p^{-1};p^{-1})_\infty}p^{-\binom{r+1}{2}}\frac{1}{(p^{-1};p^{-1})_r}.$$

If $p=2$, then

$$\lim_{n\to\infty}\mathbb{P}\big[\operatorname{rank}((S_{G_{\alpha}(n,u)})_2)=r\big]=\frac{1}{(-1;2^{-1})_\infty}2^{-\binom{r}{2}}\frac{1}{(2^{-1};2^{-1})_r}.$$

These formulas conjecture the limiting rank distributions of Sylow subgroups of sandpile groups of random bipartite graphs. The paper proves the odd-prime subgroup-distribution conjecture but does not state a resolution of this rank-distribution conjecture, so the rank formulas remain open.