Largeness of acip parameters in the quadratic family

Let the quadratic family be the parameterized family of quadratic maps, and call an invariant probability measure an acip when it is absolutely continuous with respect to Lebesgue measure. Quadratic-family acip conjecture. The set of parameters in the quadratic family which admit an acip is large. The source presents this as a deliberately informal conjecture, motivated by the lack of estimates for the size or relative proportion of such parameters. Its status is refuted according to the supplied status evidence.