  The goal of this small note is to give a more concise proof of a result due
to Berthelot, Esnault, and R\"ulling. For a regular, proper, and flat scheme
$X$ over a discrete valuation ring of mixed characteristic $(0,p)$, it relates
the vanishing of the cohomology of the structure sheaf of the generic fibre of
$X$ with the vanishing of the Witt vector cohomology of its special fibre. We
use as a critical ingredient results and constructions by Beilinson and
Nekov\'a\v{r}--Nizio{\l} related to the $h$-topos over a $p$-adic field.
