Conditional mixing conjecture for slice measures of piecewise hyperbolic maps

Let $f$ be the piecewise hyperbolic map under consideration, let $\rho$ be its SRB measure, and let $\rho_{\mathcal S}$ be the slice measure on the singular set. Write $\mathcal M$ for the phase space, $l^sY$ and $l^uY$ for the stable and unstable directional derivatives of $Y$, and let $\lambda_m$, $\mu_m$, $\mathcal S$, and $\|\cdot\|_{BV(\mathcal S)}$ have the meanings defined in the paper. For a measure $\mu$ and function $\psi$, set $\mu(\psi):=\int_{\mathbb R^2}\psi\,\mathrm d\mu$. Conditional mixing conjecture. There exists a Banach space $\mathcal B$ such that $C^1(\mathcal M)\subseteq\mathcal B\subseteq L^\infty(\mathcal M)$, $l^sY,l^uY\in\mathcal B$ for every $Y\in C^2$, and there are $c,\theta\in(0,1)$ and $C>0$ for which, for all $A,B,\Gamma$ and $m,n\geq0$, the two correlation bounds stated in the conjecture hold:

$$\left|\rho_{\mathcal S}\left((A\circ f^{n+m})(B\circ f^m)\Gamma\lambda_m^{-1}\right)-\rho(A)\rho_{\mathcal S}(B\circ f^m\Gamma\lambda_m^{-1})\right|\leq C\|A\|_{C^1}\|B\|_{\mathcal B}\|\Gamma\|_{BV(\mathcal S)}c^n\theta^m,$$

and

$$\left|\rho_{\mathcal S}\left((A\circ f^n)(B\circ f^{-m})\Gamma\mu_m\right)-\rho(A)\rho_{\mathcal S}(B\circ f^{-m}\Gamma\mu_m)\right|\leq C\|A\|_{C^1}\|B\|_{\mathcal B}\|\Gamma\|_{BV(\mathcal S)}c^n\theta^m.$$

This conjecture is a conditional mixing statement for the slice measure on the singular set, analogous to the cited conjecture and supported numerically in the source. Its resolution is not established by the supplied text.