Golden-ratio conjecture for normalized dilatations

Let $S$ be a closed surface and let $\phi$ be a pseudo-Anosov mapping class with dilatation $\lambda(\phi)$. For a fibered cohomology class, let $L(S,\phi)$ denote the corresponding normalized dilatation; in the genus-$g$ family considered in the paper, $L(S_{(1,g)},\phi_{(1,g)})=\lambda(\phi_{(1,g)})^{2g}$. The golden-ratio normalized-dilatation conjecture. The smallest accumulation point for normalized dilatations is $\gamma_0^4$, where $\gamma_0$ is the golden ratio. This is presented as a more general version of the genus-normalized conjecture and is motivated by a family whose normalized dilatations converge to $\gamma_0^4$; its resolution is not supplied here.