Madlener and Otto's plain group conjecture

Let $G$ be a group. Say that $G$ is plain if it is isomorphic to a free product of finitely many finite groups and finitely many copies of $\mathbb Z$. A finite convergent length-reducing rewriting system is a finite rewriting system that is convergent and whose rewriting rules strictly reduce word length. Madlener and Otto's conjecture. A group $G$ admits a presentation by a finite convergent length-reducing rewriting system if and only if $G$ is plain. A positive answer to the decision problem for whether such a presented group is not plain would disprove this longstanding conjecture; the paper places that decision problem in $\mathsf{NP}$, but does not resolve the conjecture.