Murre's conjecture on Chow–Künneth decompositions and the Bloch–Beilinson filtration

Let $X$ be a smooth projective variety of dimension $d$. A Chow–Künneth decomposition is a system of projectors $\{\pi^0,\ldots,\pi^{2d}\}$ inducing the Künneth decomposition on cohomology. Such a decomposition induces a descending filtration

$$F^j\operatorname{CH}^i(X):=\bigcap_{k>2i-j}\ker\bigl(\pi^k_*:\operatorname{CH}^i(X)\to\operatorname{CH}^i(X)\bigr)=\sum_{k\leq 2i-j}\operatorname{im}\bigl(\pi^k_*:\operatorname{CH}^i(X)\to\operatorname{CH}^i(X)\bigr).$$

Murre's conjecture. There exists a Chow–Künneth decomposition $\{\pi^0,\ldots,\pi^{2d}\}$, and the induced filtration satisfies $F^0\operatorname{CH}^i(X)=\operatorname{CH}^i(X)$, $F^{i+1}\operatorname{CH}^i(X)=0$, is independent of the decomposition, and has $F^1\operatorname{CH}^i(X)=\operatorname{CH}^i(X)_{\mathrm{hom}}$. These assertions form part of Murre's conjectures and are open in general.