Jacobian-kernel conjecture for codimension-two cycles

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$, with $n\geq 2$. Write $CH_{Jac,\mathbf Q}^2(X)$ for the kernel of the Abel–Jacobi map on homologically trivial codimension-two cycles, and let $CH_{alg,\mathbf Q}^2(X)$ denote the subgroup of algebraically trivial cycles. Jacobian-kernel conjecture.

$$CH_{Jac,\mathbf Q}^2(X)\subset CH_{alg,\mathbf Q}^2(X).$$

This asserts that every homologically trivial codimension-two cycle with vanishing Abel–Jacobi invariant is algebraically trivial; the source gives no resolution status.