Entropy and periodic points in pseudo-Anosov isotopy classes

Let $f$ be a pseudo-Anosov homeomorphism of a closed orientable surface of genus at least two. We prove that, for every $C^\infty$ diffeomorphism $g$ isotopic to $f$, the exponential growth rate of the number of nonempty Nielsen classes of its iterates equals $h_{\mathrm{top}}(g)$ if and only if $h_