Weak Gersten conjecture for one-relator RG groups

A restricted Gromov group, or RG group, is a finitely generated, torsion-free group $G$ such that for all $g,h\in G$, either $\langle g,h\rangle$ is cyclic or there exists some $i\in\mathbb Z$ such that $\langle g^i,h^i\rangle$ is free of rank two, and every element of $G$ is contained in a maximal cyclic subgroup of $G$. A group is one-relator if it admits a presentation with one defining relator.

Weak Gersten conjecture. Every one-relator RG group is hyperbolic.

This is presented as a weak form of Gersten's conjecture, which asserts that every one-relator group with no Baumslag--Solitar subgroups is hyperbolic. Since RG groups contain no Baumslag--Solitar subgroups, the conjecture would explain why hyperbolicity is relevant to the results while extending them beyond the hyperbolic setting. Its status is not resolved in the supplied source.