Effective SPR conjecture for surface diffeomorphisms

Let $f$ be a $C^\infty$ surface diffeomorphism with positive topological entropy. For an ergodic measure $\mu$ of $f$, write $h_\mu(f)$ for its metric entropy, $h_{\rm top}(f)$ for the topological entropy, $L^u_{\beta_0}(f)$ and $L^s_{\beta_0}(f)$ for the sets of points with unstable and stable manifolds of the prescribed uniform size, and $E^u(x)$ and $E^s(x)$ for the unstable and stable directions at $x$.

Effective SPR conjecture. For every $1>\alpha_1>\alpha_2>0$, there exist $\beta_0>0$ and $C>0$ such that, for every ergodic measure $\mu$ of $f$ satisfying

$$h_\mu(f)\geq \alpha_1 h_{\rm top}(f),$$

we have

$$\mu\Big(L^u_{\beta_0}(f)\cap L^s_{\beta_0}(f)\cap\big\{x:\angle\big(E^u(x),E^s(x)\big)>C\big\}\Big)>\alpha_2.$$

This is expected to strengthen the preceding theorem, which proves the corresponding conclusion with lower bound $2\alpha_2-1$ under the stronger assumption $\alpha_2>\tfrac12$. The conjecture asserts that the same type of uniform stable and unstable control and uniform angle separation should hold for every $\alpha_2>0$.