Chow-theoretic Chern class conjecture for Lie-irreducible 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 Lie irreducible with quasi-unipotent monodromy at infinity. Let $(V_\rho,\nabla_\rho)$ be the associated algebraic flat vector bundle. Lie-irreducible Chern class conjecture. For every $i\geq 2$,

$$r^i c_i(V_\rho)=\binom{r}{i}c_1(V_\rho)^i\quad\text{in }\operatorname{CH}^i(X)_{\mathbb Q}.$$

In particular, if the image of $\rho$ is contained in $\operatorname{SL}(r,\mathbb C)$, then $c_i(V_\rho)$ is torsion in $\operatorname{CH}^i(X)$ for every $i\geq 2$. This is suggested by results for rank two representations and remains open in general.