Persistence of nonuniform expansivity under small perturbations

Let $f$ be a nonuniformly expanding map. Perturbation-persistence conjecture. Sufficiently small perturbations of $f$ have positive probability of also being nonuniformly expanding. The conjecture concerns statistical persistence under perturbation and is motivated by the hoped-for equivalence between induced Markov maps and nonuniform expansivity. The source does not provide a resolution.