A new proof of a vanishing result due to Berthelot, Esnault, and R\"ulling

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.