Occurrence of interior spectral edges for periodic Schrödinger operators

Consider a planar embedding of a periodic graph with equal angles between the tangent lines of the edges meeting at each vertex, and let $\Omega_\varepsilon$ be the associated fattened-graph domain. Define the $\mathbb{Z}^2$-periodic Schrödinger operator

$$S=-\Delta+V(x)$$

on $\mathbb{R}^2$, where $V(x)=0$ in $\Omega_\varepsilon$ and $V(x)=C$ outside it. Interior-edge conjecture. Under appropriate asymptotics $\varepsilon\rightarrow 0$ and $C\rightarrow\infty$, the spectrum of $S$ will display an isolated spectral band whose edges are attained inside the Brillouin zone. The proposed mechanism is an explicit construction of interior spectral edges for periodic Schrödinger operators, but the source states that no rigorous argument establishing the claim is currently available; related thin-sleeve limits can fail because low-energy bound states may appear at vertices.