Marked lens rigidity conjecture for manifolds of Anosov type

Let $M$ be a smooth manifold with boundary, and let $g_1$ and $g_2$ be smooth metrics of Anosov type on $M$ such that

$$g_1|_{T(\partial M)}=g_2|_{T(\partial M)}.$$

Assume that $g_1$ and $g_2$ have the same marked lens data, meaning that their lens data agree and the corresponding geodesics are homotopic relative to their endpoints. Marked lens rigidity conjecture. There exists a smooth diffeomorphism $\psi$, homotopic to the identity and equal to the identity on $\partial M$, such that

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

This conjecture was solved in dimension $2$ and locally for pairs of negatively curved metrics in higher dimensions; the general statement for manifolds of Anosov type is therefore recorded as solved according to the supplied status.