Relative Kato homology conjecture for semilocal global models

Let $K$ be a global field, let $\mathcal{O}_S$ be a regular semilocal subring of $K$, and set $\mathcal{S}=\operatorname{Spec}(\mathcal{O}_S)$. Let $\mathcal{X}$ be a regular scheme proper and flat over $\mathcal{S}$ with smooth generic fiber $X$. Define

$$KC^{(0)}(\mathcal{X}/S,\mathbb{Z}/n\mathbb{Z})=\operatorname{coker}\left[KC^{(0)}(\mathcal{X},\mathbb{Z}/n\mathbb{Z})[1]\rightarrow\bigoplus_{v\in\sum_S}KC^{(1)}(X_{K_v},\mathbb{Z}/n\mathbb{Z})\right],$$

where $\sum_S$ is the set of places of $K$ not corresponding to closed points of $\mathcal{S}$. Let $KH_a^{(0)}(\mathcal{X}/S,\mathbb{Z}/n\mathbb{Z})$ be its degree-$a$ homology. Relative Kato homology conjecture.

$$KH_a^{(0)}(\mathcal{X}/S,\mathbb{Z}/n\mathbb{Z})=0\qquad\text{for }a>0.$$

This adds a semilocal relative vanishing statement to Kato's list and is designed for arithmetic models over semilocal bases; the supplied text gives no resolution status.