Gilman's conjecture on length-reducing rewriting systems

Let $G$ be a group. A finite convergent length-reducing rewriting system for $G$ is a presentation $(\Sigma,T)$ by such a system, where each rewriting rule has a right-hand side of length at most one. Gilman's conjecture. $G$ admits a finite convergent length-reducing rewriting system $(\Sigma,T)$ in which the right-hand side of every rule has length at most one if and only if $G$ is plain. Here, plain means that $G$ is isomorphic to a free product of finitely many factors, each factor being finite or infinite cyclic. The conjecture has been solved positively by Eisenberg and the second author, giving a characterization of the groups admitting these rewriting-system presentations.