Generic Morse-extrema conjecture for band functions

Let $V$ be a real-valued periodic potential, let $H_0=-\Delta+V$ be the associated self-adjoint operator on $\ell^2(\mathbb{Z}^d)$, and let $\lambda_m(k)$, $k\in\mathbb{R}^d$, be its band functions. Generic band-extrema conjecture. Generically, with respect to the potentials and other free parameters of the operator, the extrema of the band functions satisfy: (1) they are attained by a single band; (2) they are isolated; and (3) they are nondegenerate, meaning that their Hessians are nondegenerate. The conjecture gives a precise formulation of the expectation that band functions are generically Morse functions and is relevant to spectral-band structure and applications such as homogenization and Green's function asymptotics. Statement (1), concerning attainment by a single band, was proved; the supplied status evidence does not say that statements (2) and (3) were resolved.