  Auel-Bigazzi-B\"ohning-Graf von Bothmer proved that if a proper smooth
variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial
Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally
trivial as well. In this paper, we generalize their argument to arbitrary
unramified mod $p$ \'etale motivic cohomology groups. We also see that the
properness assumption on the variety $X$ can be dropped off by using the Suslin
homology together with a certain tame subgroup of the unramified cohomology
group.
