Generalized Bloch conjecture for correspondences on surfaces

Let $X$ be a smooth projective surface. Write $CH_0(X)_{\operatorname{hom}}$ for the group of homologically trivial zero-cycles modulo rational equivalence, and let

$$\operatorname{alb}_X:CH_0(X)_{\operatorname{hom}}\to\operatorname{Alb}(X)$$

be the Albanese map. Define

$$F^0CH_0(X)=CH_0(X),\qquad F^1CH_0(X)=CH_0(X)_{\operatorname{hom}},\qquad F^2CH_0(X)=\ker\operatorname{alb}_X\subseteq F^1CH_0(X).$$

For a $2$-cycle $Z\subseteq X\times X$, let $[Z]^*:H^2(X,\mathbb C)\to H^2(X,\mathbb C)$ be its induced cohomological action and let $Z_*:CH_0(X)\to CH_0(X)$ be its induced action on zero-cycles.

Generalized Bloch conjecture. If $[Z]^*$ vanishes on $H^0(X,\Omega^2_X)$, then $Z_*$ vanishes on $F^2CH_0(X)$.

This extends the classical Bloch conjecture from surfaces to correspondences in products of surfaces. The assertion predicts that the cohomological vanishing of a correspondence on holomorphic two-forms forces its action to vanish on the Albanese kernel of zero-cycles; it remains open in general.