The nullspace characterization of Bloch eigenvalues

Let $E$ be the linear operator in the layer-potential system for the unknown densities $[\eta;\xi]$, where $\eta=[\tau;-\sigma]$, and let $(\omega,a,b)$ denote the frequency and quasiperiodicity parameters. A nontrivial nullspace corresponds to a nontrivial field satisfying the matching, continuity, and quasi-periodicity conditions. Nullspace characterization conjecture. $(\omega,a,b)$ is a Bloch eigenvalue if and only if

$$\operatorname{Null} E \neq \{0\}.$$

This is presented as a stronger claim supported by numerical evidence, analogous to the preceding theorem, and its resolution is not established in the supplied text.