Linearized Calderón problem for the Schrödinger equation

Let $(M,g)$ be a compact Riemannian manifold with smooth boundary. Let $f \in C^{\infty}(M)$ vanish to infinite order at $\partial M$ and satisfy

$$\int_M f u_1 u_2 \,dV_g = 0$$

for all $u_j \in C^{\infty}(M)$ satisfying $\Delta_g u_j = 0$ in $M$. Linearized Calderón problem. Then $f \equiv 0$. This asks whether products of harmonic functions detect smooth functions vanishing to infinite order at the boundary, and is the injectivity question arising from linearizing the Dirichlet-to-Neumann map at $q=0$. The source presents it as the linearized form of the Calderón problem; its general status is not specified here.