Toeplitz singular Bohr–Sommerfeld conjecture

Let $\hat{J}$ and $\hat{H}$ be the semiclassical Toeplitz operators associated with a semitoric system near a focus-focus singularity. The singular Bohr–Sommerfeld formula describes the joint spectrum near the focus-focus value for semiclassical pseudodifferential operators. Toeplitz Bohr–Sommerfeld conjecture. The formula in Corollary 6.8 of Vũ Ngọc's paper holds also when the operators involved are Toeplitz operators instead of pseudodifferential operators. Such a result would provide the local spectral description needed to recover the Taylor-series invariant from the joint spectrum. The cited singular Bohr–Sommerfeld formula is available for pseudodifferential operators, whereas the operators in the coupled spin-oscillator are Toeplitz; establishing the Toeplitz analogue is therefore the missing step.