Conjectural rank distribution for Sylow subgroups of random bipartite graph sandpile groups

Let $G_{\alpha}(n,u)=G(n,\lceil\alpha n\rceil,u)$ be the Erdős–Rényi random bipartite graph, with $\frac{1}{p}<\alpha\leqslant 1$, and let $r$ be a nonnegative integer. Set $q=1/p$, and let $(a;q)_m$ denote the $q$-Pochhammer symbol.

Rank distribution conjecture. If $p$ is odd, then

$$\lim_{n\to\infty}\operatorname{Prob}\big(\operatorname{rank}((S_{G_{\alpha}(n,u)})_p)=r\big)=\mathfrak{p}_{p,0}(r)=\frac{1}{(-q;q)_{\infty}}\frac{q^{\binom{r+1}{2}}}{(q;q)_r}.$$

If $p=2$, then

$$\lim_{n\to\infty}\operatorname{Prob}\big(\operatorname{rank}((S_{G_{\alpha}(n,u)})_2)=r\big)=\mathfrak{p}_{2,1}(r)=\frac{1}{(-1;q)_{\infty}}\frac{q^{\binom{r}{2}}}{(q;q)_r}.$$

This is the rank-level consequence of the preceding Sylow subgroup distribution conjecture: the limiting rank law should agree with the corresponding distribution $\mathfrak{p}_{p,k}$, with $k=0$ for odd primes and $k=1$ for $p=2$. It is therefore open together with the distribution conjecture from which it is derived.