Bloch's correspondence conjecture for the Albanese filtration

Let $X$ be a smooth projective surface and let $Y$ be a smooth projective variety of dimension $d$. Let $A^j(-)$ denote Chow groups and let $F^\bullet$ be the filtration on zero-cycles predicted by Beilinson and Murre. Write

$$Gr^*_F\gamma:Gr^*_F A^d(Y)\longrightarrow Gr^*_F A^2(X)$$

for the map induced by a correspondence $\gamma\in A^2(Y\times X)$, and let $cl(\gamma)\in H^4(Y\times X)$ be its cohomology class.

Bloch's conjecture. For every $\gamma\in A^2(Y\times X)$, the induced action $Gr^*_F\gamma$ depends only on $cl(\gamma)$.

This is a correspondence-invariance formulation of Bloch's conjecture, linking the action of algebraic cycles on the graded Chow groups to their cohomology classes. The source derives it from Beilinson's conjectural filtration; the related Albanese-kernel consequence is known in some cases but remains open for complex general-type surfaces with $p_g=0$.