Questions on the Chow ring of complete intersections

We state several questions, and prove some partial results, about the Chow ring A⁽X) of complete intersections in projective space. For one thing, we prove that if X is a general Calabi-Yau hypersurface, the intersection product A²(X)· Aⁱ(X) is one-dimensional, for any $i>0$. We also show that quintic threefolds have a multiplicative Chow-Künneth (MCK) decomposition. We wonder whether all Calabi-Yau hypersurfaces might have an MCK decomposition, and prove this is the case conditional to a conjecture of Voisin.