The generalized Bloch conjecture for varieties with vanishing low off-diagonal Hodge numbers

Let $X$ be a smooth projective variety of dimension $m$. Assume that $H^{p,q}(X)=0$ for $p\ne q$ and $p<c$ (or $q<c$). Generalized Bloch conjecture. The cycle class map

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

is injective for $i\leq c-1$. This predicts that suitable low-dimensional Chow groups are controlled by the Hodge structure; the source records examples where the prediction is proved, but not a general resolution.