Non-abelian Hodge conjecture for local systems

Prove that any \mathbb{Z}-local system on a smooth projective complex variety X that underlies a polarizable complex variation of Hodge structure is of geometric origin.

Background

This is Simpson’s proposed non-abelian analogue of the Hodge conjecture. It predicts that integral variations of Hodge structure arise from geometry, i.e., as direct summands in the cohomology of smooth proper families, thus characterizing ‘motivic’ local systems in the non-abelian setting.