  Let $n\geq 0$ be an integer. For a normal crossing variety $Y$ over the
spectrum of a field $k$ of positive characteristic $p>0$, K.Sato defined an
\'{e}tale logarithmic Hodge-Witt sheaf $\lambda^{n}_{Y, r}$ on the \'{e}tale
site $Y_{\mathrm{\acute{e}t}}$ which agrees with $W_{r}\Omega^{n}_{Y, \log}$ in
the case where $Y$ is smooth over $\operatorname{Spec}(k)$. In this paper, we
prove the Gersten-type conjecture for $\lambda^{n}_{r}$ over the henselization
of the local ring $\mathcal{O}_{Y, y}$ of $Y$ at a point $y\in Y$. As an
application, we prove the relative version of the Gersten-type conjecture for
the $p$-adic \'{e}tale Tate twist $\mathfrak{T}_{1}(n)$ over the henselization
of the local ring $\mathcal{O}_{\mathfrak{X}, x}$ of a semistable family
$\mathfrak{X}$ over the spectrum of a discrete valuation ring $B$ of mixed
characteristic $(0, p)$ at a point $x\in \mathfrak{X}$ in the case where $B$
contains $p$-th roots of unity. Moreover, we prove a generalization of Artin's
theorem about the Brauer groups.
