Simpson's motivic conjecture for rigid semisimple representations

Let $X$ be the variety under consideration, let $k$ be an algebraically closed field of characteristic zero, and let $G=SL(n,k)$. Let $\varrho:\pi_{1}(X)\longrightarrow G$ be a rigid semisimple representation.

Simpson's motivic conjecture. The representation $\varrho\otimes k$ is a direct summand over $k$ of the monodromy representation of a motive over $X$, that is, it comes from geometry.

The paper introduces this as one of two conjectures of Simpson and discusses only partial verification. Its general status is not resolved in the supplied text.