The log Tate conjecture via homotopy K-theory

Let $s=\operatorname{Spec}(k)$ be the standard log point, let $X$ be as above, let $\ell$ be invertible in $k$, and let $m\geq 2r$. Define $KH_{i,\lim}(X)=\varinjlim_{X'}KH_i(X')$ over log modifications $X'\to X$, where $KH$ is homotopy K-theory. The log Tate conjecture via homotopy K-theory. Both maps

$$\mathbb{Q}_{\ell}\otimes\operatorname{gr}^{m-r}K_{0,\lim}(X\times\mathbb{G}_m^{m-2r})\longrightarrow H^m(X)_{\ell}(r)^G$$

and

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

are surjective. This is a proposed general-degree refinement of the log Tate conjecture; no resolution is given.