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

Let Tₙ be a $2$-dimensional determinantal hypertree on n vertices. Kahle and Newman conjectured that the p-torsion of H₁(Tₙ,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 \[P( H_1(T_n,F_2)≥ h)≥ {e⁻²⁰⁰ʰ}{(100h)⁵ʰ}.\] We also show that Tₙ is a bad cosystolic expander with positive probability.