Nori's push-forward conjecture for Chow groups

Let $X$ be a smooth projective variety over the complex numbers, let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree, and let $j:D\hookrightarrow X$ be the inclusion. Nori's push-forward conjecture. The push-forward map on rational Chow groups

$$j_*:\operatorname{CH}_k(D)\otimes \mathbb{Q}\longrightarrow \operatorname{CH}_k(X)\otimes \mathbb{Q}$$

is injective whenever $k>0$. This is proposed as the dual of the preceding Chow-theoretic Lefschetz questions; the source gives no evidence that it has been resolved.