Almost-everywhere equivalence of transitivity and LEO for beta-transformations

Let $\mathcal{T}$ be the parameter triangle for the family of beta-transformations, with parameters $(\alpha,\beta)$. A beta-transformation is topologically LEO when it has the locally eventually onto property. Transitivity–LEO equivalence hypothesis. For all beta-transformation parameter values in $\mathcal{T}$ except for countably many, topological transitivity is equivalent to topological LEO:

$$\text{topological transitivity}\Longleftrightarrow\text{topological LEO}.$$

Consequently, this equivalence holds with probability $1$: the subset of parameters in $\mathcal{T}$ for which it fails has Lebesgue measure $0$. The paper notes that the equivalence is suggested by numerical simulations, while isolated transitive non-LEO examples are known in general.