Global localization conjecture for weight-one Kato homology

Let $F$ be a global field, let $F_v$ be the completion at a place $v$, and set $X_v:=X\otimes_F F_v$. Let $p=\operatorname{ch}(F)$ and $r\geq 1$.

Global localization conjecture. For $j>0$, there is an isomorphism

$$KH_j^{(1)}(X,\mathbb{Z}/p^r)\xrightarrow{\simeq}\bigoplus_v KH_j^{(1)}(X_v,\mathbb{Z}/p^r),$$

and for $j=0$ there is a short exact sequence

$$0\to KH_0^{(1)}(X,\mathbb{Z}/p^r)\to\bigoplus_v KH_0^{(1)}(X_v,\mathbb{Z}/p^r)\to\mathbb{Z}/p^r\to0,$$

where $v$ runs over all places of $F$. This is a local-global description of the weight-one Kato homology groups.