Microlocal trace formula conjecture for the bulk-edge invariant

Let $\rho_1,\rho_2$ be the ranks of the bulk Bloch bundles $\pazocal{E}_+$ and $\pazocal{E}_-$, and assume

$$\rho_1=\rho_2=n.$$

Let $\pazocal{I}_e(P)$ denote the edge invariant, let $\mathbb{P}_h$ be the semiclassical Hamiltonian, let $f(x_1)$ be the cutoff appearing in the edge construction, and let $\mathbb{W}_h$ be the quantization of the symbol $\mathbb{W}(x,\xi)=\partial_\lambda G(x,\xi;\mathbb{P}(x_2,\xi))$, where $G$ is supported in a $2\delta$-neighborhood of the crossing set of the $n$-th and $(n+1)$-st dispersion surfaces. Microlocal trace formula conjecture. There exists $\delta_0>0$ such that, for $\delta\in(0,\delta_0)$,

$$\pazocal{I}_e(P)=\operatorname{Tr}_{\pazocal{H}}\Big(\big[\mathbb{P}_h,f(x_1)\big]\cdot\mathbb{W}_h\Big)+O(h^\infty).$$

The conjecture predicts that, in the semiclassical limit, contributions to the edge invariant are microlocalized near the wavefront set of $\mathbb{W}_h$, which lies in the region where the two relevant dispersion surfaces nearly intersect; functions microlocalized away from this region should not contribute.