Kato's homological conjecture for smooth projective varieties

Let $X$ be a connected projective smooth variety over $k$, and let $H_i^K(X,n):=H_i^{1,0}(X,n)$ denote its Kato homology with coefficients in $\mathbb{Z}/n$. Then

$$H_i^K(X,n)\simeq \begin{cases} 0 & \text{if } i\ne 0,\\ \mathbb{Z}/n & \text{if } i=0. \end{cases}$$

Kato's conjecture. The displayed isomorphism holds. This is a cohomological Hasse-principle prediction for Kato homology of smooth projective varieties over the field $k$; the supplied source does not state which cases are known or whether the conjecture has been resolved.