Eventual flow equivalence conjecture for Sturmian systems

Let $(X_{\alpha},\operatorname{\sigma}_{\alpha})$ and $(X_{\beta},\operatorname{\sigma}_{\beta})$ be Sturmian systems. Sturmian eventual-flow conjecture. Eventual flow equivalence implies topological conjugacy. The source states that this is proved for nonquadratic irrational parameters, whereas the quadratic case is posed separately as a conjecture; this statement is therefore a combined formulation whose unresolved content is the quadratic case.