  We obtain the maximal regularity for the mixed Dirichlet-conormal problem in
cylindrical domains with time-dependent separations, which is the first of its
kind. The boundary of the domain is assumed to be Reifenberg-flat and the
separation is locally sufficiently close to a Lipschitz function of $m$
variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance. We
consider solutions in both $L_p$-based Sobolev spaces and $L_{q,p}$-based
mixed-norm Sobolev spaces.
