Inverse spectral conjecture for quantum integrable systems with non-degenerate singularities

Let $\mathscr{N\!D}_n$ be the class of completely integrable systems on a $2n$-dimensional symplectic manifold with non-degenerate singularities. Let $Q(\mathscr{N\!D}_n)$ be the class of $n$-tuples $T=(T_1,\dots,T_n)$ of commuting operators whose principal symbols form an element of $\mathscr{N\!D}_n$. Inverse spectral conjecture. The asymptotics of the joint spectrum of an element $T\in Q(\mathscr{N\!D}_n)$ completely determines the symplectic manifold and the principal symbols of $T$. This refines earlier inverse spectral conjectures for integrable systems and extends the corresponding result for quantum toric systems; at the time of writing, it remains open.