The Ahlfors Measure Conjecture for mapping class group limit sets

Let $S$ be a surface, let $\operatorname{Mod}(S)$ be its mapping class group, and let $\Lambda_G$ denote the limit set of a subgroup $G\leq \operatorname{Mod}(S)$ in the projective measured lamination space $\mathbb P\mathcal{ML}(S)$. Ahlfors Measure Conjecture for mapping class groups. If $G$ is a finitely generated subgroup of $\operatorname{Mod}(S)$, then $\Lambda_G$ has zero or full measure in $\mathbb P\mathcal{ML}(S)$. This is the mapping class group analogue of the Ahlfors Measure Conjecture for Kleinian groups. While there has been substantial progress, the conjecture remains open.