The generalized Bloch conjecture for zero-cycles

Let $X$ and $Y$ be smooth projective varieties of dimension $n$, and let $Z\in CH^n(X\times Y)_{\mathbb Q}$ be a correspondence. Let $F^\bullet$ be the Bloch–Beilinson filtration and $Gr_F^\bullet$ its graded pieces. Generalized Bloch conjecture for zero-cycles. If, for some $i\leq n$, the map $[Z]^*:H^{i,0}(Y)\to H^{i,0}(X)$ vanishes, then the induced pushforward $Z_*:Gr_F^iCH_0(X)_{\mathbb Q}\to Gr_F^iCH_{m-k}(Y)_{\mathbb Q}$ also vanishes for that $i$. This formulation links holomorphic forms to the Bloch–Beilinson filtration; the source gives no resolution.