Mod $p$ sheaves on Witt flags

We characterize Cohen--Macaulay and $\varphi$-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 $\varphi$-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 $\varphi$-split special fiber.