Simple length spectrum rigidity for hyperbolic surfaces

Let $S$ and $S'$ be hyperbolic surfaces, and let the multi-set of lengths of their simple closed geodesics be called their simple length spectrum. Simple length spectrum rigidity conjecture. If $S$ and $S'$ have the same simple length spectrum, then they are isometric. This conjecture asks whether simple length isospectrality determines a hyperbolic surface up to isometry. The paper establishes non-isospectrality for a particular pair of Sunada covers, but the general question remains open.