Torsion-freeness conjecture for random cubical complex fundamental groups

Let $Q$ be a random 2-dimensional cubical complex sampled as $Q\sim Q_2(n,p)$, where $p>1-(\tfrac 12)^{1/40}$. The fundamental group $\pi_1(Q)$ is the group associated with $Q$. Torsion-freeness conjecture. With high probability, $\pi_1(Q)$ is torsion free.

This predicts the absence of finite-order elements in the fundamental group in the high-density regime for random 2-dimensional cubical complexes. The supplied context does not state whether this claim has been proved or disproved.