Homologically trivial cycles are admissible cycles

Let $X$ be a smooth projective variety over the finitely generated field $K$, and let $CH^i_l(X)$ denote the subgroup of cycles homologically equivalent to zero for $l$-adic cohomology, where $l\ne\operatorname{char} k$. Let $CH^i(X)^{(0)}$ be the subgroup of admissible cycles defined in Proposition 7. Homological admissibility conjecture. One has the inclusion

$$CH^i_l(X)\subseteq CH^i(X)^{(0)}.$$

The paper states this inclusion as a consequence of the numerical equivalence conjecture. Its general validity is therefore unresolved unless that preceding conjecture is established.