Marked length spectrum rigidity conjecture

Let $X$ be a closed manifold, and let $h_1$ and $h_2$ be Riemannian metrics on $X$ with negative sectional curvature. For each such metric, let $\mathrm{MLS}_h$ assign to every conjugacy class of $\pi_1(X)$ the length of its geodesic representative. Marked length spectrum rigidity conjecture. If

$$\mathrm{MLS}_{h_1}=\mathrm{MLS}_{h_2},$$

then $h_1$ and $h_2$ are isometric, that is, there exists a diffeomorphism $\phi:X\to X$ such that $\phi^*h_2=h_1$; equivalently, the two metrics have the same marked length spectrum if and only if they are isometric. This is a central rigidity problem in the geometry of negatively curved manifolds and is identified as an open problem of Burns and Katok.