Total expansion characterizes exponential decay of correlations

Let $f:M\to M$ be a non-uniformly expanding map with an absolutely continuous invariant measure $\mu$. The map is totally expanding when every invariant measure has all Lyapunov exponents bounded below by a common constant $\lambda>0$. Total-expansion conjecture. The map $f$ has exponential decay of correlations if and only if it is totally expanding. This proposes an equivalence between a uniform positivity condition on Lyapunov exponents across invariant measures and statistical mixing at an exponential rate; the supplied text gives motivation from one-dimensional smooth dynamics but does not state a resolution.