The closed geodesic complement conjecture

Let $N$ be a hyperbolic 3-manifold and let $c_1$ and $c_2$ be closed geodesics in $N$. Their complements are the manifolds obtained by removing the geodesics. Closed geodesic complement conjecture. Closed geodesics in a hyperbolic 3-manifold are determined by their complements, even allowing orientation-reversing homeomorphisms. The source gives this as a further conjecture after discussing cosmetic fillings and supplies no resolution.