Higher-dimensional geometric quantization conjecture for almost toric manifolds

Let $M$ be a $2m$-dimensional closed almost toric manifold with singularities of any corank and of Williamson type $(k_e,0,k_f)$, where $k_f<2$. Let $BS_r$ and $BS_{f-r}$ be the images on the base of the regular and focus-focus-regular Bohr–Sommerfeld fibers, respectively, with focus-focus-regular fibers having Williamson type $(0,0,1)$, and let $n(p)$ be the number of nodes on the fiber whose image is $p\in BS_{f-r}$. Higher-dimensional geometric quantization conjecture.

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

This extends the computation of real geometric quantization for semitoric systems to higher-dimensional almost toric manifolds. A complete symplectic topological classification near semitoric fibers in dimensions greater than four is not available, so the conjecture is based on Mayer–Vietoris and Künneth arguments and remains open.