Widom's conjecture for discontinuous pseudodifferential symbols and spectral projectors

Assume the conditions denoted by Assumptions. Let $\Pi_{\hbar,\mu}$ be the spectral projector defined in, and let

$$\Pi_\hbar|_{\Omega}=\1_{\Omega}\Pi_{\hbar,\mu}\1_{\Omega}.$$

Let $g:[0,1]\to\mathbb{R}$ be continuous with $g(0)=g(1)=0$, and suppose either $g\geq 0$ or the function

$$t\longmapsto \frac{g(t)}{t(1-t)}$$

is in $L^1$. Widom's conjecture. As $\hbar\to0$,

$$\frac{\operatorname{tr}g(\Pi_\hbar|_{\Omega})}{(2\pi\hbar)^{1-n}\log\hbar^{-1}}\longrightarrow 2C_{\Omega}\int_{[0,1]}\frac{g(\lambda)}{\lambda(1-\lambda)}\,\mathrm{d}\lambda,$$

where $C_{\Omega}$ is given by the variance formula with $f=1$. This conjecture extends Widom's asymptotic formula for discontinuous symbols to general spectral projectors; the source describes analytic cases and several partial results, while the stated extension is proposed as an open problem.