Trace asymptotics conjecture for geometric quantizations

Let $(M,\omega)$ be a closed symplectic manifold, and let $\{T_k:C^\infty(M)\to\mathcal{L}(H_k)\}_{k\in\mathbb{N}}$ be a geometric quantization satisfying the trace axiom and star product axiom. Let $\operatorname{tr}_{\hbar}$ denote the canonical trace, with $k=1/\hbar$. Trace asymptotics conjecture. For all $f\in C^\infty(M)$ and $m\in\mathbb{N}$, one has the asymptotic expansion

$$\operatorname{tr}\,T_k(f)=\operatorname{tr}_{\hbar}(f)+O(1/k^m)$$

as $k=1/\hbar\to+\infty$. The conjecture asserts agreement with the canonical trace to every order; it was established for Berezin–Toeplitz quantizations of closed Kähler manifolds by Hawkins, so the source's cited case is solved while the general statement is not resolved here.