The homology torsion growth of determinantal hypertrees

Fix a dimension $d\ge 2$, and let $T_n$ be a random $d$-dimensional determinantal hypertree on $n$ vertices. We prove that $$\frac{\log \mathrm{\mid }{H}_{d-1}(T_n,\mathbb{Z})\mathrm{\mid }}{\left(\genfrac{}{}{0px}{}{n}{d}\right)}$$ converges in probability to a constant $c_d$, which satisfies $$\frac{1}{2}\log \left(\frac{d+1}{e}\right)\le c_d\le \frac{1}{2}\log (d+1)\mathrm{.}$$