Rate-separation conjecture for partially hyperbolic toral diffeomorphisms

Let $L$ be the hyperbolic automorphism under consideration, and let $f$ be homotopic to $L$ and partially hyperbolic in the strongest sense described by the source. Denote the rate constants by $\mu_i$, $\alpha_i$, $\widetilde\alpha_i$, $\widetilde\beta_i$, $\lambda_i$, and $\beta_i$. Rate-separation conjecture. Then the rate constants satisfy

$$\mu_l<\alpha_l<\widetilde\alpha_{l-1}<\mu_{l-1}<\alpha_{l-1}<\ldots<\widetilde\beta_{k-1}<\lambda_{k-1}<\beta_{k-1}<\widetilde\beta_k<\lambda_k.$$

This is used as an alternative assumption for the foliation arguments; the authors state that they think the inequalities are automatic from the strongest partial-hyperbolicity condition, but no resolution is given.