Picard-number-one conjecture for moduli spaces with level structure

For $g\geq 3$ and $L\geq 2$, let ${\rm Mod}_g[L]$ be the level-$L$ congruence subgroup acting trivially on $H_1(\Sigma_g,\mathbb Z/L\mathbb Z)$. Picard-number-one conjecture for level $L$ structures.

$$H_2({\rm Mod}_g[L];\mathbb Q)=\mathbb Q$$

when $g\geq 3$; more generally, one should compute

$$H_2({\rm Mod}_g[L];\mathbb Z)$$

for all $g\geq 3$ and $L\geq 2$. The rational assertion would imply that the orbifold Picard group of the corresponding moduli space has rank one; even the case $(g,L)=(3,2)$ is open.