Bloch–Beilinson's correspondence conjecture for Albanese kernels

Let $S$ and $T$ be smooth projective surfaces. Let $\Gamma\in\mathrm{CH}^2(S\times T)$ be a codimension-$2$ cycle, and let

$$[\Gamma]^*:H^{2,0}(T)\longrightarrow H^{2,0}(S)$$

be the induced map. Define $\mathrm{CH}_0(S)_{\mathrm{alb}}$ and $\mathrm{CH}_0(T)_{\mathrm{alb}}$ as the kernels of their respective Albanese maps on degree-zero zero-cycles. Bloch–Beilinson's correspondence conjecture. If $[\Gamma]^*=0$, then

$$\Gamma_*:\mathrm{CH}_0(S)_{\mathrm{alb}}\longrightarrow\mathrm{CH}_0(T)_{\mathrm{alb}}$$

vanishes. This generalizes Bloch's conjecture and expresses the predicted action of correspondences on the Albanese-kernel part of the Chow group. The source gives no resolution.