Conditional mixing conjecture for analytic Anosov surface diffeomorphisms

Let $f$ be an analytic Anosov diffeomorphism of a compact surface, and let $H$ be an analytic function on the surface whose zero level set has no critical points. Conditional measures are obtained by disintegrating an invariant measure along the level curves $H=0$. Conditional mixing conjecture for analytic Anosov surface diffeomorphisms. For all analytic Anosov diffeomorphisms on compact surfaces, conditional mixing holds on an open dense set of functions $H$ with no critical points on their zero level set. This conjecture seeks to extend the robust conditional-mixing phenomenon observed for baker's maps to general analytic Anosov dynamics on compact surfaces; it remains open in the stated generality.