Subgroup tameness conjecture for hyperbolic mapping tori of free groups

Let $\mathbb{F}$ be a finitely generated free group and let $\psi\to \mathbb{F}$ be a fully irreducible monomorphism, meaning that no proper nontrivial finitely generated free factor is mapped into a conjugate of itself by a positive power of $\psi$. Set $G=M(\psi)$ and assume that $G$ is hyperbolic. For a finitely generated subgroup $H\leqslant G$, a fibre subgroup is one arising from an automorphism of a finite-index mapping-torus subgroup, and a semi-fibre subgroup is the analogous subgroup arising from a proper monomorphism. Subgroup tameness conjecture. Either $H$ is a fibre or semi-fibre subgroup of a finite-index subgroup of $G$, or $H$ is quasi-convex. This is the proposed analogue of the subgroup tameness theorem for hyperbolic 3-manifold groups; the source attributes it to an AIM workshop and Abdenbi–Wise, and gives no resolution.