The infinite-index subgroup surface-group conjecture for one-relator groups

Let $G$ be an infinite non-free one-relator group such that every subgroup of infinite index is free. A surface group is the fundamental group of a closed surface of non-positive Euler characteristic.

Infinite-index subgroup surface-group conjecture. Then $G$ is a surface group.

This is presented by the authors as their own conjecture and as substantially stronger than Surface Group Conjecture B. The source explains that it is justified by showing that it follows from Gromov's Surface Subgroup Conjecture for hyperbolic groups; its resolution status is not specified here.