Hausdorff dimension of the parameters for $(\alpha,\beta)$-transformations with the specification property

In this paper we consider the specification property for $(\alpha,\beta)$-shifts. When $\alpha=0$, Schmeling shows that the set of $\beta>1$ for which the $\beta$-shift has the specification property has the Lebesgue measure zero but has the full Hausdorff dimension\cite{Schmeling}. So it is natural to ask what happens when $\alpha>0$. Buzzi shows that for fixed $\alpha$ the set of $\beta >1$ for which the $(\alpha,\beta)$-shift has the specification property has Lebesgue measure zero. Hence we consider the Hausdorff dimension of the parameter space of $(\alpha,\beta)$-shifts.