Kahle–Newman’s Cohen–Lenstra conjecture for determinantal hypertrees

Let $T_n$ be a random $2$-dimensional determinantal hypertree on the vertex set $[n]$, and let $p$ be a prime. Write $\Gamma_{n,p}$ for the $p$-torsion subgroup of $H_1(T_n,\mathbb{Z})$. For a finite abelian $p$-group $G$, let $\operatorname{Aut}(G)$ denote its automorphism group. Kahle–Newman’s conjecture. The random group $\Gamma_{n,p}$ converges to the Cohen–Lenstra distribution: for every finite abelian $p$-group $G$,

$$\lim_{n\to\infty}\mathbb{P}(\Gamma_{n,p}\cong G)=\frac{1}{|\operatorname{Aut}(G)|}\prod_{j=1}^{\infty}(1-p^{-j}).$$

The conjecture is false for $p=2$, while it remains open for $p>2$.