The logarithmic K-theoretic Tate conjecture

Let $s$ be the standard log point over a finite field $k$, let $X$ be a projective vertical log smooth fs log scheme over $s$, and define $K_0^{\log}(X)$ as the quotient of $K_{0,\lim}(X)$ by the ideal generated by $l_a-1$ for all logarithmic line-bundle classes $l_a$. The logarithmic K-theoretic Tate conjecture.

$$\mathbb{Q}_{\ell}\otimes\operatorname{gr}^rK_0^{\log}(X)\overset{\cong}{\longrightarrow}H^{2r}(X)_{\ell}(r)^G$$

is an isomorphism. This removes the excess classes arising from logarithmic modifications and is a finer version of the degree-$2r$ log Tate conjecture; no resolution is stated.