The characterization of plain groups by finite convergent presentations

Let $G$ be a finitely generated group. A group is plain if it is a free product of free groups and finite groups.

Plain-group characterization conjecture. The following assertions are equivalent:

1. $G$ is a plain group.
2. $G$ has a finite convergent length-reducing presentation.
3. $G$ has a finite convergent weight-reducing presentation.

The conjecture compares length-reducing and weight-reducing rewriting presentations for finitely generated groups. The surrounding discussion states that the situation for groups is unclear; no resolution is supplied here.