Bhargava–de Pascale–Koenig conjecture on Sylow subgroups of random bipartite sandpile groups

Let $n_1\geq n_2$ be positive integers, let $0<u<1$, and let $G(n_1,n_2,u)$ be the Erdős–Rényi random bipartite graph whose bipartition has vertex sets of sizes $n_1$ and $n_2$, with each possible edge included independently with probability $u$. For fixed $0<\alpha\leq 1$, write $G_{\alpha}(n,u)=G(n,\lceil\alpha n\rceil,u)$, and let $S_{G_{\alpha}(n,u)}$ be its sandpile group. For a prime $p$, write $(S_{G_{\alpha}(n,u)})_p$ for its Sylow $p$-subgroup. Let $G$ be a finite abelian $p$-group. The Sylow-subgroup distribution conjecture. If $0<u<1$ and $\frac{1}{p}<\alpha\leq 1$, then

$$\lim_{n\to\infty}\mathbb{P}\big[(S_{G_{\alpha}(n,u)})_p\cong G\big]=P^{\operatorname{Sym}}_{\infty,p}(G)$$

when $p$ is odd, whereas

$$\lim_{n\to\infty}\mathbb{P}\big[(S_{G_{\alpha}(n,u)})_2\cong G\big]=\frac{|G/2G|}{2}P^{\operatorname{Sym}}_{\infty,2}(G).$$

The odd-prime assertion is proved in the paper's main theorem, while the corresponding assertion for $p=2$ remains the conjectural part. The result describes the limiting Sylow-subgroup distribution of sandpile groups of dense random bipartite graphs.