The conjectural injectivity of pushforward on Chow groups for very general ample divisors

Let $X$ be a smooth projective variety, let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree, and let $j:D\hookrightarrow X$ be the inclusion. Pushforward injectivity conjecture. The pushforward map on rational Chow groups

$$j_*:CH_k(D)\otimes \mathbb{Q}\longrightarrow CH_k(X)\otimes \mathbb{Q}$$

is injective whenever $k>0$. This is posed as the dual of the preceding Chow-theoretic Lefschetz questions and concerns the kernel of pushforward for positive-dimensional cycles. The source gives no evidence that the conjecture has been resolved.