Simon's subshift conjecture for Verblunsky coefficients

Let $\mathcal{A}$ be a finite alphabet of Verblunsky coefficients, and let $\Omega\subset \mathcal{A}^{\mathbb{Z}}$ be a minimal subshift that is not periodic. The associated measures have a common essential support. Simon's subshift conjecture. This common essential support has zero Lebesgue measure. The conjecture is motivated by the analogous question for Schrödinger operators and concerns spectral thinness for aperiodic minimal systems.