  We determine the cohomology of the closed Drinfeld stratum of
$p$-Deligne--Lusztig schemes of Coxeter type attached to arbitrary inner forms
of unramified groups over a local non-archimedean field. We prove that the
corresponding torus weight spaces are supported in exactly one cohomological
degree, and are pairwisely non-isomorphic irreducible representations of the
pro-unipotent radical of the corresponding parahoric subgroup. We also prove
that all Moy--Prasad quotients of this stratum are maximal varieties, and we
investigate the relation between the resulting representations and Kirillov's
orbit method.
