Genericity of periodic Hamiltonians

Fix a finite spectral interval $\Lambda=(a,b)$. A periodic Hamiltonian $H$ is called generic on $\Lambda$ if, for every band edge $\lambda_0$ occurring inside $\Lambda$, the band functions attain $\lambda_0$ at finitely many points of the Brillouin zone $B$, and near each such point either there is a unique band function with value $\lambda_0$ at the point and a non-degenerate extremum there, or exactly two band functions $\lambda_-$ and $\lambda_+$ meet there, satisfy $\lambda_-(k)<\lambda_+(k)$ away from the meeting point, and the function

$$D(k)=(\lambda_+(k)-\lambda_0)(\lambda_-(k)-\lambda_0)$$

has a non-degenerate maximum at the meeting point. Here non-degenerate means that the Hessian is non-degenerate. Genericity conjecture. Generic periodic Hamiltonians form a set of second Baire category in a suitable class of periodic operators. The conjecture concerns the prevalence of the stated local generic band-edge behavior among periodic operators; the source gives no resolution or precise choice of the suitable class.