Calderón's problem for the Schrödinger equation on Riemannian manifolds

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and assume that $q_1, q_2 \in C^{\infty}(M)$ so that $0$ is not a Dirichlet eigenvalue of $\Delta_g + q_j$ in $M$. Calderón problem for the Schrödinger equation. Then $\Lambda_{g,q_1} = \Lambda_{g,q_2}$ implies that $q_1 = q_2$. This is the inverse problem of determining the potential from the Dirichlet-to-Neumann map. It is proved for two-dimensional manifolds and for certain higher-dimensional settings, but remains open for general smooth metrics in dimensions at least three.