Mod p sheaves on Witt flags

We characterize Cohen--Macaulay and φ-rational perfect schemes in terms of their perverse \'etale mod p sheaves. Using inversion of adjunction, we prove that sufficiently small Schubert varieties in the Witt affine flag variety are perfections of globally $+$-regular varieties, and hence they are φ-rational. Our methods apply uniformly to all affine Schubert varieties in equicharacteristic, as well as classical Schubert varieties, thereby answering a question of Bhatt. As a corollary, we deduce that scheme-theoretic local models always have φ-split special fiber.