The asymptotic conjecture for minimal pseudo-Anosov translation lengths on punctured surfaces

Let $S_{g,n}$ be a surface of genus $g$ with $n$ punctures, let $\operatorname{Mod}(S_{g,n})$ be its mapping class group, and let $L_{\mathcal C}(\operatorname{Mod}(S_{g,n}))$ denote the minimal translation length of a pseudo-Anosov mapping class on the curve complex of $S_{g,n}$. Asymptotic translation-length conjecture. For fixed $g\geqslant 2$ and $n$ varying, $L_{\mathcal C}(\operatorname{Mod}(S_{g,n}))$ is of the order of $\frac{1}{|\chi(S_{g,n})|}$ as $n\to\infty$. The conjecture predicts that, for each fixed genus, the minimal pseudo-Anosov translation length decays like the reciprocal of the absolute Euler characteristic as the number of punctures grows. The paper gives examples yielding upper bounds of this order in some cases, including genus two, but the corresponding general asymptotic behavior is left open.