The isomorphism conjecture for homology of level mapping class groups

Let $\Sigma_g$ be a closed oriented surface of genus $g$, let $\operatorname{Mod}_g$ be its mapping class group, and let $\operatorname{Mod}_g(L)$ denote the level-$L$ mapping class group. For fixed $L \geq 2$ and $k \geq 1$, consider the natural map

$$\operatorname{H}_k(\operatorname{Mod}_g(L);{\mathbb{Q}}) \longrightarrow \operatorname{H}_k(\operatorname{Mod}_g;{\mathbb{Q}}).$$

Isomorphism conjecture. For fixed $L \geq 2$ and $k \geq 1$, there exists some $N$ such that if $g \geq N$, then this map is an isomorphism.

The conjecture predicts that, in each fixed degree, the rational homology of the level mapping class group eventually agrees with that of the full mapping class group. It is motivated by the known degree-two result and Hain's theorem on first rational homology, while the corresponding higher-degree statement remains open.