BMS conjecture on $L^2$-positivity preservation

Let $(M,g)$ be a complete Riemannian manifold. It is $L^2$-positivity preserving if, for every $u\in L^2(M)$, the implication

$$(-\Delta+1)u\geq 0 \text{ as a distribution} \Rightarrow u\geq 0$$

holds. BMS conjecture. If $(M,g)$ is geodesically complete, then $M$ is $L^2$-positivity preserving. This conjecture concerns positivity of solutions to $(-\Delta+1)u\geq0$ on complete Riemannian manifolds and is related to essential self-adjointness of Schrödinger operators. It has been proved in the Euclidean case by T. Kato; the supplied source indicates that the conjecture is resolved.