The third-band absolutely-continuous-spectrum conjecture

For even $\kappa\geq6$, define $J_3(\kappa)=(\cos(2\pi/\kappa),\cos^2(\pi/\kappa))$, and let $\boldsymbol{\mu}_{\kappa}(D)$ denote the absolutely-continuous-spectrum set defined in the paper. Third-band conjecture.

$$J_3(\kappa)\subset\boldsymbol{\mu}_{\kappa}(D).$$

This is motivated by analogy with the corresponding Laplacian and is presented without proof; the paper says that this region was only briefly investigated.