The BMS conjecture

I explain an open conjecture by Braverman/Milatovic/Shubin (BMS) on the positivity of square integrable solutions $f$ of $(-\mathrm{\Delta }+1)f\ge 0$ on a geodescially complete Riemannian manifold, and its connection to essential self-adjointness problems of covariant Schr"odinger operators. The latter conjecture has remained open for more than 14 years now.