The affine sieve conjecture for perfect Zariski closures

Let $\Gamma$ be a finitely generated subgroup of $GL_n(F)$, where $F$ is a field, and suppose $\Gamma\subset GL_n(R)$ for a finitely generated domain $R$. Let $H$ be the Zariski closure of $\Gamma$ and let $H^0$ be its connected component; assume that $H^0$ is perfect. For a finite-index ideal $I$ of $R$, define

$$\Gamma(I)=\ker\big(\Gamma\to GL_n(R/I)\big).$$

Perfect-closure property $(\tau)$ conjecture. The group $\Gamma$ has property $(\tau)$ with respect to the family $\{\Gamma(I)\}$ as $I$ ranges over all finite-index ideals of $R$. The source presents this as a speculative ultimate generalization of the square-free congruence-quotient theorem, with no proof or resolution supplied.