The log Tate conjecture in degree twice the codimension

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $\ell$ be invertible in $k$, and define

$$K_{0,\lim}(X)=\varinjlim_{X'}K_0(X'),$$

where $X'$ ranges over the log modifications of $X$; let $\operatorname{gr}^r$ denote the gamma filtration and write $H^m(X)_{\ell}=H^m_{\operatorname{\log\acute{e}t}}(X_{\overline{s}(\log)},\mathbb{Q}_{\ell})$. The log Tate conjecture. The Chern class map

$$\mathbb{Q}_{\ell}\otimes\operatorname{gr}^rK_{0,\lim}(X)\longrightarrow H^{2r}(X)_{\ell}(r)^G$$

is surjective. This is the degree-$2r$ logarithmic analogue of the classical Tate conjecture; the paper notes that it was discussed previously but does not state a resolution.