Classify two-generator one-relator groups with free finitely generated infinite-index subgroups

Determine whether a two-generator, one-relator group G in which every finitely generated subgroup of infinite index is free must be isomorphic to either Z or a solvable Baumslag–Solitar group BS(1,n).

Background

Theorem D shows that for infinite one-relator groups, either the group is free or a surface group, or it has an infinite-index subgroup that is not free; however, in the two-generator case all finitely generated infinite-index subgroups may be free (e.g., BS(1,2)).

Remark 0.3 highlights a mismatch: in two-generator cases, finitely generated infinite-index subgroups can all be free even when the group contains an infinitely generated non-free subgroup, prompting this classification question.