Spectral-band conjecture for the one-dimensional Bernoulli displacement model

Consider the one-dimensional Bernoulli displacement model with coupling parameter $\lambda$, almost-sure spectrum $\Sigma_{\lambda}$, and operators $h_{\omega,\lambda}$. Let $\omega^{\ast}$ be the distinguished displacement configuration appearing in the model, and let $\omega^1$ be the displacement configuration defined by

$$\omega^1_k=1\quad\text{for all }k\in\mathbb Z.$$

Spectral-band conjecture. For every $\lambda\neq 0$,

$$\Sigma_{\lambda}=\sigma(h_{\omega^{\ast},\lambda})\cup\sigma(h_{\omega^1,\lambda}).$$

If true, this would give the proposed description of the spectrum for all nonzero coupling parameters; in particular, for $|\lambda|>2$ it would imply the band structure discussed in the surrounding remarks. The claim is presented as conjectural and is supported by numerical evidence for values such as $\lambda=3$.