The finite-group generalization of the random group-equation threshold theorem

Let $G$ be a finite group of order at least two, and let $H_n(\pi,k,m)$ be the random uniquely extendable constraint system arising from the group-equation model described in the source. Finite-group threshold conjecture. Theorem~ should continue to hold when the assumption that $G$ is abelian is replaced by the assumption that $G$ is a finite group of order at least two. The conjecture removes commutativity from the group underlying the random constraint system and asks whether the same threshold and conclusions remain valid.