Let $S$ be a smooth connected variety over the complex numbers. The fundamental group might not be residually finite (i.e., the homomorphism $\pi_1(S(\mathbb C)) \to \pi_1^{\mathrm{et}}(S)$ might not be injective).

Is there a dense Zariski open subset $U\subset S$ such that $\pi_1(U(\mathbb C)) $ is residually finite?