Bloch–Beilinson's filtration conjecture for higher Chow groups

Let $X$ be a smooth projective variety over a field $k$, and let $C\!H^p(X,q)_{\mathbb Q}$ denote the rational higher Chow group. Let $F^\bullet$ be a filtration on these groups, with $C\!H^p(X)_{\mathrm{hom}\,\mathbb Q}$ the subgroup homologically trivial for a fixed Weil cohomology $H^\bullet(X)$.

Bloch–Beilinson filtration conjecture. There exists a filtration

$$\cdots\subset F^i\subset\cdots\subset F^1\subset F^0=C\!H^p(X,q)_{\mathbb Q}$$

satisfying: (a) $F^1C\!H^p(X,0)_{\mathbb Q}=C\!H^p(X)_{\mathrm{hom}\,\mathbb Q}$; (b) $F^rC\!H^p(X,q)_{\mathbb Q}\cdot F^sC\!H^{p'}(X,q')_{\mathbb Q}\subset F^{r+s}C\!H^{p+p'}(X,q+q')_{\mathbb Q}$ under the intersection product; (c) $F^\bullet$ is respected by $f^*$ and $f_*$ for morphisms $f:X\to Y$; and (d) $F^iC\!H^p(X,q)_{\mathbb Q}=0$ for $i\gg0$.

This is proposed as a generalisation of Bloch–Beilinson conjectures for ordinary Chow groups to higher Chow groups. Its status is open.