Saito–Sato conjecture on Kato homology over a henselian discrete valuation ring

Let $R$ be an excellent henselian discrete valuation ring with residue field $k$, and let $B=\operatorname{Spec}R$. Let $\mathcal Y$ be a regular scheme, flat and proper over $B$, and let $l$ be a prime invertible on $R$. Write $KH_a(\mathcal Y,{{\mathbb Q}}_l/{{\mathbb Z}}_l)$ for its Kato homology, and let $I$ be the number of irreducible components of the special fiber $\mathcal Y\times_B k$. Saito–Sato's conjecture. 1) If $k$ is separably closed, then

$$KH_a(\mathcal Y,{{\mathbb Q}}_l/{{\mathbb Z}}_l)=0$$

for all $a$. 2) If $k$ is finite, then

$$KH_a(\mathcal Y,{{\mathbb Q}}_l/{{\mathbb Z}}_l)=\begin{cases}0&\text{for }a\not=1;\\({{\mathbb Q}}_l/{{\mathbb Z}}_l)^I&\text{for }a=1. \end{cases}$$

This describes the Kato homology groups in the separably closed and finite residue-field cases; the source attributes these conjectures to Saito and Sato and uses a strengthening of them in the local argument.