Bloch's generalized conjecture on Chow groups and Hodge coniveau

Let $X$ be a smooth projective variety of dimension $m$ satisfying

$$H^{p,q}(X)=0$$

for $p\ne q$ and $p<c$ or $q<c$. The cycle map is

$$cl: CH_i(X)_{\mathbb{Q}}\rightarrow H^{2m-2i}(X,\mathbb{Q}).$$

Bloch's generalized conjecture. For every integer $i\leq c-1$, the cycle map $cl$ is injective. This is presented as a converse to the theorem that injectivity of low-dimensional cycle maps forces the stated Hodge vanishing; it is open in general and generalizes Bloch's conjecture for surfaces.