Kahle–Newman Cohen–Lenstra conjecture for random hypertree torsion

Let $p$ be an odd prime, let $T_n$ be the random hypertree in the paper, and let $\Gamma_{n,p}$ denote the $p^\infty$-torsion subgroup of $H_1(T_n,\mathbb{Z})$. For a finite abelian group $G$, write $\operatorname{Aut}(G)$ for its automorphism group.

Kahle–Newman conjecture. For every finite abelian $p$-group $G$,

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

This is the Cohen–Lenstra limiting distribution. The source says the stronger version is false for $p=2$, while the odd-prime case remains open; it would imply the stated limiting distribution for the mod-$p$ first Betti number.