Generic structure conjecture for spectral edges of periodic operators

Let $\lambda_j(k)$ denote the band functions of a periodic operator, and consider their extrema with respect to the quasimomentum $k$. A property is called generic when it holds for potentials and other free parameters outside the exceptional choices.

Generic spectral-edge conjecture. Generically, the extrema of band functions are attained by a single band, are isolated, and are non-degenerate, meaning that their Hessians are non-degenerate.

Equivalently, near a generic spectral edge the dispersion relation should have a single parabolic shape, resembling the dispersion relation at the bottom of the spectrum of $-\Delta$. The source notes that each of the listed complications can occur, but presents their generic absence as an unresolved general belief.