The nonabelian Hodge conjecture for motivic representations

Let $X_s$ be a smooth projective fiber of a smooth projective family $X\to S$, and let $NL(X_s,GL(n))$ denote the locus of integral, equivalently $\mathbb Z$-, variations of Hodge structure on $X_s$. A motivic representation means a representation arising from motivic data.

Nonabelian Hodge conjecture. The points of $NL(X_s,GL(n))$, that is, the $\mathbb Z$-variations of Hodge structure, are motivic representations on $X_s$.

This is presented as a nonabelian analogue of the Hodge conjecture and as a proposed explanation for the quasiprojective algebraicity of the nonabelian Noether–Lefschetz locus. Its status is open in the source.