Rayleigh-quotient monotonicity for spectral mass

Let $\Delta$ and $\Delta'$ be two Laplacians whose Rayleigh quotients satisfy

$$\mathcal{R}(u)\leq\mathcal{R}'(u)\qquad\text{for all }u.$$

For each sign, let $M^{\pm}(\lambda)$ and $(M^{\pm})'(\lambda)$ denote the corresponding upper and lower asymptotic spectral masses at threshold $\lambda$. Rayleigh-quotient monotonicity conjecture. Then

$$M^{\pm}(\lambda)\geq (M^{\pm})'(\lambda).$$

This would permit Rayleigh-quotient comparisons to yield spectral-mass estimates. The source notes that it should be easy to prove when an integrated-density-of-states definition is available and that it follows from the subsequent conjecture.