The Beauville–Laszlo uniformization conjecture for $G$-bundles

Let $E$ be the local field in the paper, let $\operatorname{Gr}$ be the corresponding affine Grassmannian, and let

$$\mathcal{BL}:\operatorname{Gr}\longrightarrow \operatorname{Bun}_G\times_{\operatorname{Spa}(\mathbb{F}_q)}\operatorname{Spa}(E)^\diamond$$

be the Beauville–Laszlo morphism. Beauville–Laszlo uniformization conjecture. The morphism $\mathcal{BL}$ is surjective locally for the smooth topology: for every $S\in\operatorname{Perf}_{\mathbb{F}_q}$ and every element of $(\operatorname{Bun}_G\times\operatorname{Spa}(E)^\diamond)(S)$, there is a smooth surjective cover $\widetilde S\to S$ over which that element comes from an element of $\operatorname{Gr}(\widetilde S)$. This strengthens the known surjectivity on geometric points and would provide the desired local uniformization of the moduli stack.