  Let $X$ be a cubic hypersurface in $\mathbb P^6$ or a hypersurface of degree
greater than equal to $7$ in $\mathbb P^5$. In this note we try to understand,
for a very general hyperplane section of $X$, the non-injectivity locus of the
corresponding push-forward homomorphism at the level of Chow group of certain
dimension.
