Kato's conjecture for the Kato complex of 0-cycles

Let $S$ be the spectrum of a finite field or a henselian discrete valuation ring with finite residue field, and let $X$ be a smooth projective scheme over $S$. For an integer $n$ invertible on $X$, write $KH^{(0)}_a(X)$ for the homology in degree $a$ of the Kato complex of 0-cycles. Kato's conjecture.

$$KH^{(0)}_a(X)=\begin{cases} \mathbb Z/n\mathbb Z & \text{if $a=0$ and $\dim S=0$},\\ 0 & \text{otherwise}. \end{cases}$$

This is the 0-cycle analogue of higher-dimensional class field theory and the cohomological Hasse principle. The source presents it as the framework motivating the paper, but supplies no resolution status here.