Kato's localization conjecture for Kato homology over global fields

Let $F$ be a global field of characteristic $p\geq 0$, let $l\neq p$ be a prime, and for each place $v$ let $F_v$ be the completion of $F$ at $v$. Let $X$ be the scheme under consideration and set $X_v:=X\otimes_F F_v$. Let $KH_j^{(1)}(-,\mathbb{Z}/l^r)$ denote degree-$j$ Kato homology with coefficients in $\mathbb{Z}/l^r$. Kato's localization conjecture. For $j>0$, there is an isomorphism

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

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

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

The statement is a local-global assertion for Kato homology over a global field; the supplied text gives no resolution status.