Nori's Chow-theoretic Lefschetz conjecture

Let $X$ be a smooth projective variety over the complex numbers, and let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree. Write $j:D\hookrightarrow X$ for the inclusion. Nori's conjecture. The restriction map

$$j^*:\operatorname{CH}^p(X)\otimes \mathbb{Q}\longrightarrow \operatorname{CH}^p(D)\otimes \mathbb{Q}$$

is an isomorphism for $p<\dim D$ and is injective for $p=\dim D$. This is a Chow-theoretic analogue of the Lefschetz hyperplane theorem; the source attributes it to M. Nori, and no resolution status is given here.