Statistical stability of physical measures in $\mathcal{U}$

Let $\mathcal{U}$ be the open family of maps considered in the paper, and let $f_n$ be $C^\infty$ dynamics in $\mathcal{U}$ with physical measures $\mu_n$. Suppose that $f_n$ converges to $f\in\mathcal{U}$ in a $C^r$ topology, where $2\leq r\leq\infty$, and let weak convergence refer to convergence of probability measures. Statistical stability conjecture. The physical measures of the $C^\infty$ dynamics in $\mathcal{U}$ are statistically stable: whenever $\mu_n$ is the physical measure for $f_n$, the sequence $\mu_n$ weakly converges to an $f$-invariant probability measure $\mu$ that is also a physical measure for $f$. Statistical stability would describe the continuity of physical measures under perturbations within the open family, conditional in the source on positive answers to the preceding conjectures.