Bloch's conjecture on zero-cycles and vanishing holomorphic forms

Let $X$ be a smooth, projective variety over $\mathbb{C}$, and write $h^{p,0}(X)$ for the Hodge number measuring global holomorphic $p$-forms. Bloch's conjecture. If

$$h^{p,0}(X)=0$$

for every $p\geq 1$, then

$$\operatorname{CH}_0(X)\cong \mathbb{Z}.$$

This is the converse to the implication from triviality of the zero-cycle group to vanishing holomorphic forms; it is known in several cases but remains open in general.