  In this paper, we investigate, for varieties over $\mathbb C$ with trivial
group of $0$-cycles, the gap between essential $\mathrm{CH}_0$-dimension $2$
and essential $\mathrm{CH}_0$-dimension $0$. In particular, we present
sufficient (and necessary) conditions for a variety with trivial group of
$0$-cycles and essential $\mathrm{CH}_0$-dimension $\leq 2$ to have, in fact,
essential $\mathrm{CH}_0$-dimension $0$.
