Linial–Meshulam torsion-freeness conjecture for first homology

Let $Y_d(n,p)$ denote the Linial–Meshulam random $d$-dimensional simplicial complex, and let $c_2$ be the constant appearing in the homological phase transition. A sequence $p=p(n)$ has $|np-c_2|$ bounded away from $0$ if there is a positive constant $\varepsilon$ such that $|np-c_2|\geq \varepsilon$ for all sufficiently large $n$. Torsion-freeness conjecture. For every such $p=p(n)$, $H_{1}(Y_d(n,p);\mathbb{Z})$ is torsion-free with high probability. The conjecture concerns the apparent absence of torsion in first homology away from the critical window around $c_2$; extensive experiments support it, but the statement is presented as unproved.