Nori's conjecture for Chow groups of a general hypersurface

Let $X$ be a smooth projective variety, let $L$ be a very ample line bundle on $X$, and let $S=\mathbb{P}(H^0(X,L^n)^*)$ parametrize hypersurfaces defined by sections of $L^n$. Write $K$ for the function field of $S$, let $\overline{K}$ be its algebraic closure, and let $\mathcal{Y}_K\subset\mathcal{X}_K=X\times\operatorname{Spec}(K)$ be the generic hypersurface. The inclusion induces a restriction map on Chow groups.

Nori's conjecture. If $n$ is sufficiently large, then the natural map

$$CH^i(\mathcal{X}_{\overline{K}})\otimes\mathbb{Q}\longrightarrow CH^i(\mathcal{Y}_{\overline{K}})\otimes\mathbb{Q}$$

is an isomorphism for $i<\dim(\mathcal{Y}_K)$ and an inclusion for $i=\dim(\mathcal{Y}_K)$.

The conjecture predicts that, in the indicated codimensions, a sufficiently general high-degree hypersurface acquires no new rational Chow classes beyond those coming from the ambient variety, with the boundary codimension allowing injectivity rather than surjectivity.