Finiteness conjecture for the componentwise Albanese kernel

Let $X=C_1\times\cdots\times C_d$ be a product of smooth projective curves over a number field $k$, and let $F^2(X)$ be the kernel of the Albanese map on degree-zero zero-cycles. Define $F^2(X)_{\mathrm{comp}}$ to be the subgroup generated by elements of $F^2(X)$ expressible as $z_1\times\cdots\times z_d$, where each $z_i$ is a zero-cycle on $C_i$. Componentwise finiteness conjecture. The group $F^2(X)_{\mathrm{comp}}$ is finite. This is presented as a weaker consequence of the Bloch–Beilinson conjecture for products of curves. The supplied text does not give a resolution, so the assertion remains open here.