The motivic cohomology and K-theory comparison conjecture

For every variety $X$ over ${\mathbb{Q}}$, suppose there is a universal cohomology theory $X\mapsto h^*(X)$ with values in an abelian category ${\mathcal M}{\mathcal M}$ of mixed motives, together with a universal absolute cohomology theory whose motivic cohomology groups are denoted $H_{\mathcal M}^n(X,j)$. The associated Leray spectral sequence should give short exact sequences

$$0\to \operatorname{Ext}^1_{{\mathcal M}{\mathcal M}}({\mathbb{Q}}(0),h^{n-1}(X)(j))\to H_{\mathcal M}^n(X,j)\to \operatorname{Hom}_{{\mathcal M}{\mathcal M}}({\mathbb{Q}}(0),h^n(X)(j))\to 0.$$

For smooth $X$, write $K_m(X)_{\mathbb{Q}}$ for rational K-theory and $\operatorname{Gr}^j_\gamma K_m(X)_{\mathbb{Q}}$ for its $j$-th gamma-graded piece. Beilinson et. al.'s motivic comparison conjecture. For smooth varieties $X$, there should be natural isomorphisms

$$H_{\mathcal M}^i(X,j)\cong \operatorname{Gr}^j_\gamma K_{2j-i}(X)_{\mathbb{Q}}.$$

This is part of the motivic philosophy relating absolute motivic cohomology to K-theory and the cohomology of mixed motives. The source presents it as a general expectation and gives no resolution status.