Asymptotic translation lengths for mapping class groups of punctured surfaces

Let $S_{g,n}$ be a genus-$g$ surface with $n$ punctures, let $\Mod(S_{g,n})$ be its mapping class group, let $\PMod(S_{g,n})$ be the pure mapping class group, and let $L_{\C}(G)$ denote the least asymptotic translation length on the curve complex among elements of a group $G$. Write $\chi(S_{g,n})$ for the Euler characteristic. The asymptotic translation-length conjecture. The followings hold.

1. $L_{\C}(\PMod(S_{g,n})) \asymp 1/|\chi(S_{g,n})|$.
2. $L_{\C}(\Mod(S_{g,n})) \asymp 1/|\chi(S_{g,n})|^2$.

These two assertions would extend the distinct asymptotic behaviors known in related settings to punctured surfaces. The first is supported by results for fixed genus, while the second is proposed as a general phenomenon; the source does not provide a resolution.