Pelayo-Vũ Ngọc inverse spectral conjecture for semitoric systems

A semitoric system is an integrable system $F=(J,H)$, with semiclassical joint spectrum given by the points $(\lambda,\nu)\in\mathbb R^2$ where $\lambda$ is an eigenvalue of $\hat J_\hbar$ and $\nu$ is an eigenvalue of $\hat H_\hbar$ on the corresponding $\lambda$-eigenspace as $\hbar\to0$. Two semitoric systems are symplectically equivalent when related by a symplectic equivalence. Pelayo-Vũ Ngọc conjecture. A semitoric system is determined, up to symplectic equivalence, by its semiclassical joint spectrum; moreover, from any such semiclassical spectrum one can explicitly construct the associated semitoric system. The statement is presented as an inverse problem motivated by the classification of semitoric systems, and the source does not give a resolution status.