The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra

Let $T_n$ be a $2$-dimensional determinantal hypertree on $n$ vertices. Kahle and Newman conjectured that the $p$-torsion of $H_1(T_n,\mathbb{Z})$ asymptotically follows the Cohen-Lenstra distribution. For $p=2$, we disprove this conjecture by showing that given a positive integer $h$, for all large enough $n$, we have $$\mathbb{P}(\dim H_1(T_n,\mathbb{F}_2)\ge h)\ge \frac{e^{-200h}}{(100h)^{5h}}.$$ We also show that $T_n$ is a bad cosystolic expander with positive probability.