The weak stability conjecture for level mapping class groups

Let $\Sigma_g$ be a closed oriented surface of genus $g$, let $\operatorname{Mod}_g(L)$ be the level-$L$ mapping class group, and let $\gamma$ be a simple closed nonseparating curve on $\Sigma_g$. Write $(\operatorname{Mod}_g(L))_\gamma$ for the stabilizer of $\gamma$ in $\operatorname{Mod}_g(L)$. For fixed $L \geq 2$ and $k \geq 1$, consider the natural map

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

Stability conjecture. For fixed $L \geq 2$ and $k \geq 1$, there exists some $N$ such that if $g \geq N$ and $\gamma$ is a simple closed nonseparating curve on $\Sigma_g$, then this map is a surjection.

This weaker stability statement is presented as sufficient for the isomorphism conjecture, extending the degree-two homological argument to higher degrees. Its general validity is not established in the source.