Flexibility conjecture for the entropy of (a,b)-continued fraction transformations

Let ${\mathcal P}$ be the parameter space of pairs $(a,b)$, let $f_{a,b}$ denote the corresponding $(a,b)$-continued fraction transformation, and let $h_{\rm top}(f_{a,b})$ denote its topological entropy. Let $\kappa$ be the spectral radius of

$$M_{-1,0}=\begin{pmatrix}1&1&0&0\\0&0&0&1\\0&1&0&0\\0&0&1&1\end{pmatrix},$$

so that $\kappa^3-\kappa^2-1=0$. Flexibility conjecture. (i) If $(a,b)\in{\mathcal P}$, then

$$\log(\kappa)\le h_{\rm top}(f_{a,b})\le\log\left(\frac{1+\sqrt5}{2}\right).$$

(ii) For every $h\in[\log(\kappa),\log((1+\sqrt5)/2)]$, there exists $(a,b)\in{\mathcal P}$ such that $h_{\rm top}(f_{a,b})=h$. This conjecture proposes both global bounds for the entropy and that every value between them is attained; the cited context presents these as unanswered questions supported by numerical tests.