Generalized Bloch conjecture on support of zero-cycles

Let $X$ be a smooth projective complex variety and let $m$ be a nonnegative integer. Say that $\operatorname{CH}_0(X)$ is supported on an $m$-dimensional algebraic subset $V\subset X$ when the pushforward map $\operatorname{CH}_0(V)\to\operatorname{CH}_0(X)$ is surjective. Generalized Bloch conjecture. If

$$H^{p,0}(X)=0\text{ for all }p>m,$$

then $\operatorname{CH}_0(X)$ is supported on an $m$-dimensional algebraic subset of $X$. This generalizes Bloch's conjecture by replacing the condition that all positive-degree holomorphic forms vanish with vanishing above degree $m$; the source presents it as the relevant generalization for the paper, without giving resolution evidence.