Torsion-freeness conjecture for homology of random simplicial complexes

Let $Y_d(n,p)$ be the random $d$-dimensional simplicial complex, and let $c_d^*$ be the real number such that $p=c_d^*/n$ is a one-sided sharp threshold for the non-triviality of $H_d(Y_d(n,p);\mathbb R)$. For $p=p(n)$, suppose that $|np-c_d^*|$ is bounded away from $0$. Torsion-freeness conjecture. For every $d\geq 2$, the group $H_{d-1}(Y_d(n,p);\mathbb Z)$ is torsion-free asymptotically almost surely. The conjecture predicts that torsion does not appear in random simplicial complexes in this regime; the source does not give a resolution.