Super approximation conjecture for finitely generated subgroups of $\operatorname{SL}_n(\mathbb Z)$

Let $\Gamma<\operatorname{SL}_n(\mathbb Z)$ be finitely generated, and let $\mathbb G$ be the Zariski closure of $\Gamma$, with identity component $\mathbb G_0$. The group $\mathbb G_0$ is perfect when

$$[\mathbb G_0,\mathbb G_0]=\mathbb G_0.$$

Super approximation conjecture. The group $\Gamma$ has the super approximation property with respect to all positive integers if and only if $\mathbb G_0$ is perfect.