The stable-direction tail dependence conjecture for mixing rates

Consider smooth diffeomorphisms with hyperbolic physical measures. A stable leaf has uniform size on a full-volume subset if there is a cylinder in the ambient space on which that size is uniform. Stable-direction tail dependence conjecture. There are examples of smooth diffeomorphisms, with hyperbolic physical measures, whose stable leaves admit no cylinder where their size is uniform, on a full-volume subset, and whose mixing rates depend on the tail of hyperbolic times along the stable direction, namely the analogue of the hyperbolic-time tail subset with $\phi^{cs}$ in place of $\phi^{cu}$. This proposes that, when uniform stable-leaf geometry is absent, the stable-direction hyperbolic-time tail can govern the rate of mixing rather than only the unstable-direction tail.