The generalized Bloch conjecture for smooth hypersurfaces

Let $Y\subset\mathbb{P}^{n}$ be a smooth hypersurface of degree $d$. Write $A_i(Y):=CH_i(Y)_{\mathbb{Q}}$ for the Chow group of $i$-dimensional cycles with rational coefficients modulo rational equivalence, and let $A_i^{hom}(Y)\subset A_i(Y)$ be the subgroup of homologically trivial cycles. The generalized Bloch conjecture. One has

$$A_i^{hom}(Y)=0\qquad\forall\ i\leq \frac{n}{d}-1.$$

This prediction expresses the influence of the Hodge level of the cohomology on the size of the Chow groups; in the surface case it is the still-open Bloch conjecture. The stated hypersurface prediction remains open, although partial results are known.