The hullizability conjecture for joint spectra of commuting semiclassical unitary operators

Consider pairwise commuting unitary semiclassical operators $U_1(\hbar), \ldots, U_d(\hbar)$ with joint principal symbol $F=(f_0^1,\ldots,f_0^d)$, and assume that $F(M)$ is closed. The joint spectrum is denoted by $\mathrm{JointSpec}(U_1(\hbar),\ldots,U_d(\hbar))$. Hullizability conjecture. Assume that $F(M)$ is hullizable. From the behaviour of the joint spectrum $\mathrm{JointSpec}(U_1(\hbar),\ldots,U_d(\hbar))$ as $\hbar$ tends to zero, one can recover the convex hull of $F(M)$. This proposes extending the preceding spectral recovery result beyond the assumption that the principal symbols are surjective; the precise meaning of hullizability is given in the source's Definition of the paper.