Semiclassical Weyl law for the Baouendi–Grushin example

On $\mathbb{T}^2$, let $X=\partial_x$, $Y=2(1-\cos x)\partial_y$, and let $\Delta_H=X^2+Y^2$. The singular set is $S=\{0\}\times\mathbb{T}$, and $c>0$ is the constant in the singular Weyl measure $c\delta_S$. Semiclassical Weyl-law conjecture for the Baouendi–Grushin example. For $q>0$ and every potential $V\in C(\mathbb{T}^2,\mathbb{R})$,

$$\lim_{h\rightarrow 0^+} h^3N^{-}\big(h^{2q}\Delta_H^q+V\big) = c\int_{\mathbb{T}} V_{-}(0,y)^{\frac{3}{2q}}\,dy.$$

This is motivated by the singular sR-dimension $3$ and the corresponding singular Weyl measure; the source presents it as a conjecture and gives no resolution.