Density of uniformly exponential decay in generic one-parameter families

Consider a generic one-parameter family of maps. Density conjecture. Maps with exponential decay of correlations are Lebesgue density points of other maps with uniform exponential rates of decay of correlations, and maps with at least polynomial decay are Lebesgue density points of maps with arbitrarily small exponential decay. The source attributes related results for Benedicks–Carleson families to Tsujii, but the stated density claim is presented as a conjectural principle.