The coherence conjecture of Pappas and Rapoport, proved by Zhu, establishes an equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a specific union of Schubert varieties in a partial affine flag variety. In this talk, I will discuss a refinement of the coherence conjecture by endowing line bundles on global Schubert varieties with an equivariant structure under the ramified global jet group scheme. This refinement has implications for the study of affine Demazure modules. The talk is based on joint work with Huanhuan Yu.