Monotonicity conjecture for Schrödinger operators under geometric flows

Let $S$ be a Schrödinger operator evolving with a geometric flow $\mathcal{F}$. Let $\Delta$ denote the Laplace–Beltrami operator, and suppose that the eigenvalues of $-\Delta$ are non-decreasing under $\mathcal{F}$. Monotonicity conjecture. Then the eigenvalues of $S$ are also non-decreasing under $\mathcal{F}$. The conjecture proposes that spectral monotonicity for Schrödinger operators is inherited from that of the Laplace–Beltrami operator; the source presents empirical evidence from several geometric operators, but gives no resolution in general.