The locally quasiconvex virtual mapping-torus conjecture

Let $\mathcal{G}$ be a locally quasiconvex hyperbolic group. An immersion of finite directed height is an immersion $\psi:H\rightarrow F$ of graphs satisfying the finite directed-height condition used in the source. The locally quasiconvex virtual mapping-torus conjecture. If $\mathcal{G}$ is a locally quasiconvex hyperbolic group, then $\mathcal{G}$ has a finite index subgroup isomorphic to the fundamental group of a mapping torus of an immersion $\psi:H\rightarrow F$ of finite directed height. This is a structural conjecture up to finite index; the source does not report a resolution.