The finer logarithmic 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 let $E$ be a log Tate curve. Define $K(X,m,r)$ as the subgroup of $\mathbb{Q}\otimes\operatorname{gr}^{m-r}K_0^{\log}(X\times E^{m-2r})$ consisting of elements satisfying the pullback and symmetric-group invariance conditions stated in the source, and set $K(X,m,r)=0$ for $m<2r$. The finer logarithmic Tate conjecture.

$$\mathbb{Q}_{\ell}\otimes K(X,m,r)\overset{\cong}{\longrightarrow}H^m(X)_{\ell}(r)^G$$

is an isomorphism. This refines the log Tate conjecture in all degrees $m\geq 2r$ and is motivated by weight-monodromy; the source gives no resolution.