Adiabatic asymptotic formula for arbitrary foliations

Let ${\mathcal F}$ be an arbitrary foliation on a compact Riemannian manifold. In the above notation, let $\Delta_F$ and $\Delta_H$ denote the leafwise and transverse Laplacians, respectively, let $\Delta_{{\mathcal F}_N}$ be the corresponding leafwise Laplacian for the foliation ${\mathcal F}_N$ on $N^*{\mathcal F}$, and let $g_N$ denote the corresponding transverse symbol term. For any function $f\in C^\infty_c({\mathbb R})$, the adiabatic asymptotic conjecture. The asymptotic formula holds:

$$\operatorname{tr} f(\Delta_F + h^2\Delta_H) =(2\pi)^{-q}h^{-q} \operatorname{tr}_{{\mathcal F}_N} f(\Delta_{{\mathcal F}_N}+g_N) +o(h^{-q}),\quad h\rightarrow 0.$$

The formula is known for Riemannian foliations and is conjectured to remain valid for arbitrary foliations; establishing it would extend the adiabatic spectral asymptotics beyond the Riemannian setting.