Quantitative spectral-projector bound for quantum completely integrable manifolds

Let $M$ be a compact smooth manifold of dimension $d$ that is quantum completely integrable, let $\Sigma$ be the associated hypersurface, and assume that $\Sigma$ is degenerate at most of order $N\geq2$: at every point of $\Sigma$, there is a unit-speed curve in $\Sigma$ whose derivative of some order between $2$ and $N$ is nonzero. Let $\omega$ be the set of points $x\in M$ for which $T_x^*M\cap\Omega$ has codimension at most $1$, and let $K\subset\omega$ be compact.

Quantum complete integrability conjecture. There exists $\kappa=\kappa(N,d)>0$ such that

$$\delta\geq\lambda^{-\kappa}\quad\Longrightarrow\quad \\|P_{\lambda,\delta}\\|_{L^2(M)\to L^\infty(K)}\lesssim \lambda^{(d-1)/2}\delta^{1/2}.$$

This would extend the surface-of-revolution method to general quantum completely integrable manifolds with finite-type degeneracy. The required parametrices and the asserted estimate are presented as expected future results, so the conjecture remains open.