Mel'nikov's conjecture on residually finite one-relator groups

A finitely generated group $G$ is Mel'nikov if all its subgroups of finite index are one-relator groups.

Mel'nikov's conjecture. An infinite and residually finite Mel'nikov group is either a free group, a surface group or a solvable Baumslag--Solitar group.

This problem is cited in the source as an open problem, and the paper presents its results as progress towards it. The conjecture remains open in general, although the two-generated case is proved by Gardam, Kielak and Logan.