Griffiths's conjecture on incidentally trivial cycles

Let $X$ be a nice variety over an algebraically closed field $K$ of dimension $d$, and let $I^p(X)$ denote the subgroup of incidentally trivial codimension-$p$ cycles inside $\operatorname{CH}^p_{\mathrm{alg}}(X)$. Let $\mathrm{AJ}^p$ be the Abel--Jacobi map.

Griffiths's conjecture on incidentally trivial cycles. For every $p\in\{1,\ldots,d\}$ and every $z\in I^p(X)$, there is a positive integer $N$ such that

$$Nz\in\ker(\mathrm{AJ}^p).$$

Equivalently, incidentally trivial cycles are Abel--Jacobi trivial up to torsion; the source says this is equivalent to the assertion that the associated surjection $h^p$ is an isogeny. The supplied text gives no resolution of the conjecture.