Semiclassical Weyl law for flat quantum tori

Let $\mathbb{T}^n_\theta$ be the quantum torus, let $\Delta$ be its flat Laplacian, let $q>0$, set $r=2nq^{-1}$, and suppose that either $r\neq 1$ and $r'=\max(r,1)$, or $r=1<r'$. For a selfadjoint potential $V=V^*\in L_{r'}(\mathbb{T}^n_\theta)$, let $N(h^{2q}\Delta^q+V;\lambda)$ be the counting function at energy $\lambda\in\mathbb{R}$. Semiclassical Weyl-law conjecture for flat quantum tori. For every $\lambda\in\mathbb{R}$,

$$\lim_{h\rightarrow 0^+} h^nN\big(h^{2q}\Delta^q+V;\lambda\big) = \hat{c}(n)\tau_0\big[(V-\lambda)_{-}^{\frac{n}{2q}}\big], \qquad \hat{c}(n):=|\mathbb{B}^n|.$$

This was presented as a conjectured semiclassical Weyl law for flat quantum tori; the source gives no resolution.