Torsion conjecture for Chern classes of properly rigid representations

Let $X$ be a smooth complex quasi-projective variety with base point $x$, and let

$$\rho:\pi_1(X^{\mathrm{an}},x)\to\operatorname{GL}(r,\mathbb C)$$

be a properly rigid representation with quasi-unipotent monodromy at infinity. Let $(V_\rho,\nabla_\rho)$ be its associated algebraic flat vector bundle. Torsion conjecture for properly rigid representations. For every $i\geq 1$, $c_i(V_\rho)$ is torsion in $\operatorname{CH}^i(X)$.

The conjecture is motivated by the vanishing of analytic and secondary Chern classes for such representations. It would follow from the geometric-origin conjecture for rigid local systems together with the corresponding torsion statement for flat bundles of geometric origin, but remains open in general.