Pappas–Rapoport's conformal-blocks conjecture for parahoric bundle stacks

Let $X$ be a smooth projective curve, let $S\subseteq X$ be a nonempty set containing the bad points $\mathcal{R}$ of a parahoric Bruhat–Tits group scheme $\mathcal{G}$, and let $\operatorname{Bun}_{\mathcal{G}}$ be the moduli stack of $\mathcal{G}$-bundles. For each $x\in S$, let $\operatorname{Gr}_{\mathcal{G},x}$ be the affine Grassmannian and let $q_x\colon \operatorname{Gr}_{\mathcal{G},x}\to\operatorname{Bun}_{\mathcal{G}}$ be the uniformization map. For a Lie algebra $Y$ acting on a vector space $V$, write $[V]^Y$ for the elements of $V$ annihilated by $Y$. Let $\mathcal{L}$ be a dominant line bundle on $\operatorname{Bun}_{\mathcal{G}}$. Pappas–Rapoport's conjecture. There is a canonical isomorphism

$$\mathrm{H}^0(\operatorname{Bun}_{\mathcal{G}},\mathcal{L})\cong\left[\bigotimes_{x\in S}\mathrm{H}^0(\operatorname{Gr}_{\mathcal{G},x},q_x^*\mathcal{L})\right]^{\mathrm{H}^0(\mathring{X},\operatorname{Lie}(\mathcal{G}))}.$$

Here $\mathring{X}=X\setminus S$. This is the precise representation-theoretic description of sections referred to in the introduction; the paper presents it as a statement suggested by Pappas and Rapoport, and its resolution status is not established by the supplied material.