On a theorem of Scott and Swarup

Let $1 \rightarrow H \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1$ be an exact sequence of hyperbolic groups induced by an automorphism $\phi$ of the free group $H$. Let $H_1 ( \subset H)$ be a finitely generated distorted subgroup of $G$. Then there exist $N > 0$ and a free factor $K$ of $H$ such that the conjugacy class of $K$ is preserved by $\phi^N$ and $H_1$ contains a finite index subgroup of a conjugate of $K$. This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.