Generic gap-edge nondegeneracy conjecture for periodic elliptic operators

Let $P$ be a generic selfadjoint second-order elliptic operator with periodic coefficients, and let $(\lambda_{-},\lambda_{+})$ be a nontrivial gap in its spectrum.

Generic gap-edge conjecture. Each endpoint of the gap is a unique, modulo the dual lattice, nondegenerate extremum of a single band function $\lambda_j(k)$.

This conjecture concerns the analytic structure of the dispersion relation near the Fermi surface, which is needed for detailed Liouville theorems and dimension formulas for polynomially growing solutions. The source presents it as an expected statement and gives no resolution.