Continuity and monotonicity 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. Continuity and monotonicity conjecture. (i) The function

$$(a,b)\longmapsto h_{\rm top}(f_{a,b})$$

is continuous. (ii) For fixed $b\le\tfrac12$, the function

$$a\longmapsto h_{\rm top}(f_{a,b})$$

is monotone non-decreasing. These properties would provide global regularity and a parameter monotonicity principle for the entropy; the source presents them as conjectural questions based on numerical and Markov-partition calculations.