Entropy plateau conjecture for (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. Entropy plateau conjecture. If $b\le\tfrac12$ and $-1\le a\le-1/(b+1)$, then

$$h_{\rm top}(f_{a,b})=h_{\rm top}(f_{-1,b}).$$

Equivalently, $h_{\rm top}(f_{a,b})$ is independent of $a$ in the region

$$\{(a,b)\in{\mathcal P}\mid b\le\tfrac12,\ -1\le a\le-1/(b+1)\}.$$

This predicts the horizontal entropy plateaus observed in the numerical graph, with the flat section for each fixed $b$ occurring on the indicated interval; its status is not resolved in the supplied material.