The virtual mapping-torus conjecture for locally quasiconvex hyperbolic groups

A group is locally quasiconvex if every finitely generated subgroup is quasiconvex, and a group is hyperbolic in the sense of Gromov. A fully irreducible partial endomorphism is a monomorphism from a free factor of a free group satisfying the fully irreducibility condition described in the source. The virtual mapping-torus conjecture. Let $\mathcal{G}$ be a locally quasiconvex hyperbolic group. Then $\mathcal{G}$ has a finite index subgroup $\mathcal{G}'$ such that $\mathcal{G}'=\pi_1X$, where $X$ is a mapping torus of a fully irreducible partial endomorphism of a free group. The claim proposes a structural characterization up to finite index; the source presents it as conjectural and gives no resolution.