Uniform lower bound for proper normal subgroups of mapping class groups

Let $S_{g,n}$ be a genus-$g$ surface with $n$ punctures, let $\Mod(S_{g,n})$ be its mapping class group, and let $L_C(H)$ denote the least asymptotic translation length on the curve complex among elements of a subgroup $H$. Write $\chi(S_{g,n})$ for the Euler characteristic. Uniform lower-bound conjecture. There exists a uniform constant $C>0$ such that for any proper normal subgroup $H$ of $\Mod(S_{g,n})$,

$$L_C(H) \geq \frac{C}{|\chi(S_{g,n})|}.$$

This would give a uniform lower bound for the least asymptotic translation length of every proper normal subgroup and support the distinction between mapping class groups and their proper normal subgroups on punctured surfaces. The source proposes the statement as an open problem, with partial evidence from the Torelli subgroup.