The free-subgroup conjecture for non-orientable cyclic presentations

Let $P_2(w)$ be a non-orientable cyclic presentation and let $G_2(w)$ be its associated group. Free-subgroup conjecture. Either $G_2(w)$ contains a non-abelian free subgroup, or $G_2(w)$ is isomorphic to one of $\mathbb{Z}$, $\mathbb{Z}^2$, or $BS(1,-1)$. This conjecture proposes a classification of the exceptional non-orientable two-generator cyclically presented groups; the preceding results establish it in several families, but the general case remains open.