Kato's residue conjecture for local fields

Let $F$ be a local field with finite residue field $k$, let $\mathcal{O}_F$ be its valuation ring, and let $\mathscr{X}$ be a regular scheme proper and flat over $\mathcal{O}_F$, with special fiber $X_k$. Let $X$ denote the generic fiber, and let $KH_j^{(n)}(-,\mathbb{Z}/l^r)$ be the degree-$j$ Kato homology group with coefficients in $\mathbb{Z}/l^r$. The residue map is

$$\Delta_j: KH_j^{(1)}(X,\mathbb{Z}/l^r)\to KH_j^{(0)}(X_k,\mathbb{Z}/l^r).$$

Kato's residue conjecture. The map $\Delta_j$ is an isomorphism for all $r\geq 1$ and $j\geq 0$. This conjecture concerns comparison of Kato homology on a regular proper flat model over a local field with that of its special fiber; the supplied text gives no resolution status.