Chow–Künneth decomposition conjecture

Let $k$ be an algebraically closed field, let $X$ be a smooth projective variety of dimension $d$ over $k$, and let $H^*$ be a Weil cohomology. Write the Künneth decomposition of the diagonal as

$$\Delta_X=\sum_{j=0}^{2d}\Delta_{j,X}\in H^{2d}(X\times X).$$

Chow–Künneth conjecture. There exist correspondences $\pi_{j,X}\in CH^d(X\times X)$ satisfying

$$\pi_{j,X}^2=\pi_{j,X},\qquad \pi_{j,X}\circ\pi_{j',X}=0\quad\text{for }j\ne j',$$ $$\sum_j\pi_{j,X}=\Delta_X,$$

and $cl(\pi_{j,X})=\Delta_{j,X}$ for any choice of Weil cohomology.

Such algebraic mutually orthogonal idempotent lifts of the Künneth components would give a motivic decomposition of $X$ and strengthen Grothendieck's standard conjecture that the Künneth components are algebraic. The source does not state whether this stronger conjecture is resolved.