Irreducibility conjecture for Fermi surfaces of periodic Schrödinger operators

Let $H=-\Delta+V(x)$ be a periodic Schrödinger operator, let $\Gamma^*$ be its reciprocal lattice, and let $F_{\lambda,H}$ be the real Fermi surface at energy $\lambda$, defined by

$$F_{\lambda,H}=\{k\in\mathbb{R}^n\mid H(k)u=\lambda u\text{ has a non-zero solution}\}.$$

The Fermi surface is considered modulo $\Gamma^*$-shifts.

Fermi-surface irreducibility conjecture. For $\lambda\in\mathbb{R}$, the Fermi surface $F_{\lambda,H}$ is irreducible modulo $\Gamma^*$, possibly except for a discrete set of values of $\lambda$.

Fermi surfaces are periodic level sets of the dispersion relation, and their irreducibility is closely related to the global analytic structure of periodic Schrödinger operators. The source gives no resolution of this assertion.