Madlener–Otto 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. Madlener–Otto conjecture. $G$ admits a finite convergent length-reducing rewriting system $(\Sigma,T)$ 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. This conjecture is presented as part of the program to characterize algebraically the groups admitting presentations by length-reducing systems; its resolution status is not specified in the supplied text.