  We prove an unconditional (but slightly weakened) version of the main result
of our earlier paper with the same title, which was, starting from dimension
$4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety
with trivial Chow groups, (i.e. the cycle class map to cohomology is injective
on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample
hypersurface $Y$ in $X$ is ``parameterized by cycles of dimension $c$'', then
the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.
