The existence of TGS links in closed orientable 3-manifolds

Let $M$ be a closed orientable $3$-manifold. A TGS link is a link $K$ embedded in a $3$-manifold such that $M\setminus K$ is hyperbolic and $K$ has at least one totally geodesic spanning surface. TGS-link existence conjecture. Every closed orientable $3$-manifold contains at least one TGS link. This would show that totally geodesic spanning surfaces of knots and links occur broadly among closed orientable $3$-manifolds; the supplied text gives motivation but does not report a proof or disproof.