The universality conjecture for irrational two-step affine systems

Let ${\boldsymbol{u}}$ denote the relevant dynamical system, let $A_{2,\alpha}$ be the associated system parametrized by an irrational parameter $\alpha$, let ${\mathcal{F}}(A_{2,\alpha})$ denote its characteristic class, and let ${\mathrm{\mathbf{DISP}}}_{\mathrm{e}c}$ denote the class of ergodic-disjoint systems. Universality conjecture.

$$\bigcap_{\alpha\notin\mathcal{Q}}{\mathcal{F}}(A_{2,\alpha})={\mathrm{\mathbf{DISP}}}_{\mathrm{e}c}.$$

This asserts that the systems belonging to every class ${\mathcal{F}}(A_{2,\alpha})$ for irrational $\alpha$ are exactly the ergodic-disjoint systems. The surrounding questions concern universal models for these classes; the source does not provide a resolution of this conjecture.