Compact K\'ahler manifolds with compactifiable universal cover

Let X be a compact K\"ahler manifold such that the universal cover admits a compactification. We conjecture that the fundamental group is almost abelian and reduce it to a classical conjecture of Iitaka.
Comment: With an appendix by Fr\'ed\'eric Campana (entitled "Abelianity, Iitaka and S conjectures"). 17 pages, no figure