Equality of admissible and numerically trivial Chow cycles

Let $X$ be as above, and let $CH^i(X)^{(0)}$ be the subgroup defined in Proposition 7. Let $CH^i_{\operatorname{num}}(X)$ denote the subgroup of cycles numerically equivalent to $0$. Numerical equivalence conjecture. In Proposition 7, one has

$$CH^i(X)^{(0)}=CH^i_{\operatorname{num}}(X).$$

This is the numerical analogue of the conjecture attributed to Beilinson. If $X$ has a semistable model, it predicts that $CH^i_{\operatorname{num}}(X)$ is the saturation of $CH^i(X)^0$ in $CH^i(X)$, and the paper reduces the assertion to the semistable case with $k$ perfect. The general assertion remains open.