Braverman–Milatovic–Shubin conjecture on the $L^2$-positivity preserving property

Let $(M,g)$ be a Riemannian manifold. It has the $L^2$-positivity preserving property if every $u\feature in L^2(M)$ satisfying

$$(-\Delta+1)u\geq 0$$

in the sense of distributions is non-negative almost everywhere.

Braverman–Milatovic–Shubin conjecture. If $(M,g)$ is a geodesically complete Riemannian manifold, then the $L^2$-positivity preserving property holds.

The conjecture connects positivity preservation for the Schrödinger operator $-\Delta+1$ with essential self-adjointness of Schrödinger operators with non-negative locally square-integrable potentials. The source gives no evidence of a resolution.