The star-system hyperbolicity conjecture

A star system is a dynamical system for which all sufficiently small perturbations have only hyperbolic closed orbits; its non-wandering set is denoted by $\Omega$. Star-system hyperbolicity conjecture. Does a star system have its non-wandering set hyperbolic? This question asks whether the hyperbolicity of all closed orbits robustly forces hyperbolicity of the entire non-wandering set. In the conservative setting, the non-wandering set is the whole manifold, so the question is closely related to the structural stability conjecture.