  In their study of local models of Shimura varieties for totally ramified
extensions, Pappas and Rapoport posed a conjecture about the reducedness of a
certain subscheme of $n \times n$ matrices. We give a positive answer to their
conjecture in full generality. Our main ideas follow naturally from two of our
previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai,
Magyar, and Weyman on the equations defining type A affine Grassmannians. The
second is the work of the first two authors and Kamnitzer on affine
Grassmannian slices and their reduced scheme structure. We also present a
version of our argument that is almost completely elementary: the only
non-elementary ingredient is the Frobenius splitting of Schubert varieties.
