Geometric quantization conjecture for non-degenerate integrable systems with hyperbolic components

Let $M$ be a closed symplectic manifold equipped with a non-degenerate integrable system whose singularities have any corank and Williamson type $(k_e,k_h,h_f)$, with $k_h<2$, $k_f<2$, and $k_h+k_f\leq 1$. Let $BS_r$, $BS_{f-r}$, and $BS_{h-r}$ denote the images on the base of the regular, focus-focus-regular, and hyperbolic-regular Bohr–Sommerfeld fibers, respectively, and let $n(p)$ denote the number of nodes on the fiber over $p\in BS_{f-r}$. Hyperbolic geometric quantization conjecture.

$$\mathcal{Q}(M)\cong\left(\bigoplus_{p\in BS_r}\mathds{C}\right)\oplus\left(\bigoplus_{p\in BS_{f-r}}\oplus_{n(p)} C^\infty(\mathds{R};\mathds{C})\right)\oplus\left(\bigoplus_{p\in BS_{h-r}}\mathds{C}^{\mathds{N}}\oplus\mathds{C}^{\mathds{N}}\right).$$

This proposed extension incorporates the infinite-dimensional contributions associated with hyperbolic components, alongside the regular and focus-focus contributions. The source points to existing results for singular Lagrangian fibrations with hyperbolic components, but gives no resolution of this higher-dimensional claim.