Torsion conjecture for Chern classes of flat bundles of geometric origin

Let $X$ be a smooth complex quasi-projective variety and let $\mathbb V$ be a local system of geometric origin on $X$. Let $(V,\nabla)$ be the associated algebraic flat vector bundle. Geometric-origin torsion conjecture. All Chern classes of $(V,\nabla)$ are torsion in the Chow ring:

$$c_i(V)\text{ is torsion in }\operatorname{CH}^i(X)\quad\text{for every }i\geq 1.$$

This is presented as a strengthening of Esnault's conjecture on Chern classes of Gauss–Manin connections and would imply the properly rigid case assuming Simpson's geometric-origin conjecture. It remains open in general.