The log Tate conjecture for a log Tate curve

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, let $m\geq 2r$, and let $E$ be a log Tate curve over $s$. Write $H^m(X)_{\ell}=H^m_{\operatorname{\log\acute{e}t}}(X_{\overline{s}(\log)},\mathbb{Q}_{\ell})$. The log Tate curve conjecture. The map

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

is surjective. The finite-field hypothesis makes the rational K-theory and log étale cohomology independent of the choices defining the log Tate curve; the source gives no resolution.