Irreducibility conjecture for periodic Schrödinger Bloch varieties

Let $L=-\Delta+V(x)$ be a periodic Schrödinger operator, or more generally a second-order elliptic periodic operator, with Bloch variety $\mathcal{B}_L$. Bloch-variety irreducibility conjecture. The Bloch variety $\mathcal{B}_L$ is irreducible. In dimension two this is stated as a theorem for continuous periodic potentials; the conjecture asks for the corresponding assertion for arbitrary periodic Schrödinger operators and even for general second-order elliptic periodic operators. Irreducibility is significant because it rules out flat-band components associated with bound states and implies strong analytic-continuation rigidity of the dispersion relation.