Structural stability implies hyperbolicity for rational maps

Let $f$ be a rational map. It is structurally stable if, within a holomorphic family containing it, all sufficiently nearby maps are topologically conjugate to it; it is hyperbolic if its critical points tend under forward iteration to attracting periodic cycles, equivalently if its Julia set admits a neighborhood with a uniformly expanding conformal metric. Structural-stability conjecture. A structurally stable rational map is hyperbolic. The paper's abstract notes this implication for structurally stable rational maps with Cantor Julia sets, while the general assertion is presented as a conjectural consequence of the density problem.