Lens rigidity conjecture for manifolds of Anosov type

Let $(M_1,g_1)$ and $(M_2,g_2)$ be smooth Riemannian manifolds of Anosov type whose boundary metrics agree:

$$(\partial M_1,g_1|_{\partial M_1})=(\partial M_2,g_2|_{\partial M_2}).$$

Let $(\ell_g,S_g)$ denote the lens data. Lens rigidity conjecture. If

$$(\ell_{g_1},S_{g_1})=(\ell_{g_2},S_{g_2}),$$

then there exists a smooth diffeomorphism $\psi$, equal to the identity on the boundary, such that

$$\psi^*g_2=g_1.$$

This extends boundary and lens rigidity from negatively curved or simple manifolds to the broader class of manifolds of Anosov type. The source states that there are no counterexamples to this conjecture and identifies lens rigidity in this class as open.