Shafarevich conjecture for varieties with large fundamental group

Let $X$ be a smooth complex projective variety. Its fundamental group is large if, for every closed irreducible positive-dimensional subvariety $Z\subset X$, the image

$$\operatorname{Im}\bigl[\pi_1(Z^{\mathrm{norm}})\longrightarrow\pi_1(X)\bigr]$$

is infinite. Shafarevich conjecture. If $X$ has a large fundamental group, then its universal covering $\widetilde{X}$ is Stein. This is a partial converse to the fact that Stein universal covers contain no positive-dimensional compact subvarieties. The source states that the conjecture remains open in full generality, although the linear case is known.