Let $M$ be a hyperbolic 3-manifold with finitely generated fundamental group. Assume $E$ is a geometrically infinite end (not of geometrically finite type, i.e. the convex core can not be separated of the end for a small neighborhood of $E$). Then there exists a sequence of closed geodesics approaching $E$. Call this family $\mathcal{F}$.

Question: Is it true that there are closed geodesics (in the above family) that are arbitrarily far apart? To be more precise: for a closed geodesic $\gamma$ define $l_\gamma$ as the minimum distance from $\gamma$ to a geodesic in $\mathcal{F}$. Is it true that $\{l_\gamma\}_{\gamma\in\mathcal{F}}$ is unbounded?

If the answer is no, do people know hypothesis that could imply such a statement?.