Geisser's finiteness conjecture for Kato homology with $\mathbb Q/\mathbb Z$ coefficients

Let $X$ be smooth and projective over a finite field, and let $H_i^K(X,\mathbb Q/\mathbb Z(1))$ denote Kato homology in weight $1$. Write $CH_1(X)_{\mathbb Q}=CH_1(X)\otimes\mathbb Q$, and let $\operatorname{corank}$ denote the corank of a torsion group. Kato-homology finiteness conjecture in weight $1$. The group $H_i^K(X,\mathbb Q/\mathbb Z(1))$ is finite for every $i\ne 2$, while

$$\operatorname{corank} H_2^K(X,\mathbb Q/\mathbb Z(1))=\dim CH_1(X)_{\mathbb Q}.$$

The conjecture predicts precise finiteness except in the degree governed by the rational one-cycle group; its general status is open.