The polynomial decomposition criterion for holomorphic forms

Let $X$ be a smooth projective variety. A degree $l$ polynomial decomposition of the diagonal means a decomposition of the diagonal whose non-supported part lies in the subalgebra of $\operatorname{CH}^\bullet(X\times X)$ generated in degree at most $l$. Polynomial decomposition criterion. $X$ admits a degree $l$ polynomial decomposition of the diagonal if and only if the algebra $H^0(X,\Omega^\bullet)$ is generated in degree at most $l$. This conjectural equivalence would connect a decomposition of the diagonal, an algebraic-cycle condition, with generation of the holomorphic-form algebra.