Bloch's conjecture on Chern classes of flat vector bundles

Let $X$ be a smooth complex projective variety. Let $(V,\nabla)$ be a flat vector bundle on $X$, and let

$$0=F^{i+1}\subset F^i\subset\dots\subset F^1\subset F^0=\operatorname{CH}^i(X)_{\mathbb Q}$$

be the conjectural functorial Bloch–Beilinson filtration. Bloch's conjecture. For every $i\geq 1$,

$$c_i(V)\otimes\mathbb Q\in F^i\operatorname{CH}^i(X)_{\mathbb Q}.$$

This is a precise form of the prediction that Chern classes of flat vector bundles lie in the lowest stratum of the Bloch–Beilinson filtration. The conjecture is known for $i=1$ and, by Reznikov's theorem, for $i=2$ under the expected descriptions of the first filtration steps; it remains open in higher codimensions.