The virtual 2-step nilpotency conjecture for fundamental groups of special varieties

Let $X$ be a smooth special quasi-projective variety. Its fundamental group $\pi_1(X)$ is virtually nilpotent of class at most $2$, meaning that a finite-index subgroup of $\pi_1(X)$ is nilpotent of nilpotency class at most $2$.

2-step nilpotency conjecture. The group $\pi_1(X)$ is virtually nilpotent of class at most $2$.

This refines the previously stated virtual nilpotency conjecture for smooth special quasi-projective varieties. The source provides no resolution status for this refined conjecture.