Transitivity conjecture for beta-transformations in the parameter triangle

Let $\mathcal{T}$ be the parameter triangle for the family of beta-transformations, with parameters $(\alpha,\beta)$. Transitivity conjecture. For all parameters in $\mathcal{T}$ satisfying

$$\bigl(\beta\le\sqrt{2}\ \text{and}\ \alpha\le1-1/\beta\bigr)$$

or

$$\bigl(\beta\le\sqrt{2}\ \text{and}\ \alpha\ge1+1/\beta-\beta\bigr),$$

i.e. belonging to the union of the two blue curvilinear triangles restricted by the red curves, the corresponding beta-transformations are topologically transitive. Numerical simulations indicate that this extends the rigorously known transitivity regions below $\beta=\sqrt{2}$, but the claim has not yet been proved.