Residue invariance of weight-one Kato homology under regular proper models

Let $F$ be a local field of characteristic $p>0$ with finite residue field $k$, let $X$ be a smooth variety over $F$, and let $\mathscr{X}$ be a model of $X$ over the valuation ring $\mathcal{O}_F$. Write $X_k$ for its special fiber. For $r\geq 1$ and $j\geq 0$, let $\Delta_j$ denote the residue map from weight-one Kato homology of $X$ to weight-zero Kato homology of $X_k$.

Residue invariance conjecture. If the model $\mathscr{X}$ is proper over $\mathcal{O}_F$ and regular, then

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

is an isomorphism for all $r\geq 1$ and $j\geq 0$.