The Bloch–Lewis conjecture on algebraically trivial codimension-two cycles

Let $X$ be a smooth variety. Write $CH^2(X)_{\emph{alg}}$ for the group of algebraically trivial codimension-two cycles, and let $J_a^2(X)$ be the abelian variety given by the image of the restriction of the Abel–Jacobi map to $CH^2(X)_{\emph{alg}}$. Assume

$$H^{2,0}(X)=0,$$

and assume that the generalized Hodge conjecture holds for $X$.

Bloch–Lewis conjecture. Then $CH^2(X)_{\emph{alg}}$ is finite-dimensional, and consequently

$$CH^2(X)_{\emph{alg}}\cong J_a^2(X).$$

Here finite-dimensionality means that a smooth curve and a correspondence from it induce a surjection onto $CH^2(X)_{\emph{alg}}$. The claim connects the generalized Hodge conjecture with the structure of algebraically trivial codimension-two cycles and their Abel–Jacobi images. The source gives no resolution status for this implication.