Improved eigenvalue asymptotics conjecture for critical-order operators on Carnot manifolds

Let $(M,H)$ be a compact Carnot manifold, and let $T\in \Psi^{-d_H}_H(M)$ be a pseudodifferential operator on $(M,H)$ in the van Erp-Yuncken sense. Let $\nu$ be a density on $M$, identify $T$ with its realisation as an operator on $L_2(M,\nu)$, and assume that $T$ is self-adjoint with respect to the inner product on $\nu$. Improved eigenvalue asymptotics conjecture. There exists $\delta>0$ such that, as $n\to\infty$, we have

$$\lambda(n,T)=\frac{\mathrm{Res}(T)}{d_H}n^{-1}+O(n^{-1-\delta}).$$

This conjecture seeks a remainder estimate improving the basic Weyl-type asymptotic for eigenvalues of critical-order pseudodifferential operators in the trivially filtered case. The supplied context states that the analogous asymptotic without an explicit remainder follows from the main theorem, while the stronger remainder estimate is presented as an open problem.