The projectivity conjecture for fundamental groups of compact Kähler manifolds

Let $X$ be a compact Kähler manifold. Its fundamental group is denoted by $cpi_1(X)$.

Projectivity conjecture. The fundamental group $cpi_1(X)$ is projective, meaning that there exists a projective manifold $M$ such that

$$cpi_1(X) csimeq cpi_1(M).$$

The conjecture asks whether fundamental groups of compact Kähler manifolds can always be realized by projective manifolds. In the paper's setting, the claim is proved for compact Kähler threefolds, while the general-dimensional statement remains open.