Schrödinger subshift conjecture

Let $\mathcal{A}\subset \mathbb{R}$ be finite, and let $\Omega\subset \mathcal{A}^{\mathbb{Z}}$ be a minimal subshift that is not periodic. By minimality, the Schrödinger operators generated by elements of $\Omega$ have a common spectrum. Schrödinger subshift conjecture. The associated common spectrum has zero Lebesgue measure. This conjecture is the Schrödinger-operator analogue of Simon's subshift conjecture and concerns whether aperiodic minimal finite-alphabet systems have spectrally negligible common spectrum.