Finite-height subgroups of hyperbolic groups are quasi-convex

Let $G$ be a hyperbolic group, and let $H$ be a finitely generated subgroup of $G$. Say that $H$ has finite height if there is a bound on the number of essentially distinct conjugates of $H$ with infinite intersection.

Finite-height subgroup conjecture. If $H$ has finite height in $G$, then $H$ is quasi-convex in $G$.

This conjecture concerns the converse to the fact that quasi-convex subgroups have finite height, and would characterise quasi-convexity among finitely generated subgroups of hyperbolic groups by this finiteness property. The supplied text gives no evidence that it has been resolved.