An influential conjecture of Finkelberg–Mirkovic identifies the regular block of representations of a reductive group in positive characteristic with Iwahori constructible perverse sheaves on the affine Grassmannian. We will review the statement and history of this conjecture, including its recent resolution by Bezrukavnikov–Riche. We will then discuss a second conjecture, made at the same time by Finkelberg–Mirkovic, which identifies the regular block of representations of a Frobenius kernel with Iwahori constructible perverse sheaves on the semi-infinite flag variety, as well as its proof, which is work in progress with Achar, Riche, and Taylor.