Betti Hecke-functor compatibility conjecture

Let $G$ be a complex reductive group, let $X$ be a smooth complex projective curve, and let $Gr_G=G(\mathcal K)/G(\mathcal O)$ be the affine Grassmannian, where $\mathcal K=\mathbb C((t))$ and $\mathcal O=\mathbb C[[t]]$. Let $\mathcal Shv_c(G(\mathcal O)\backslash Gr_G)$ be the Satake category and let $\mathcal Shv_{\mathcal N}(Bun_G(X))$ be the category of nilpotent sheaves. Betti Hecke-functor compatibility conjecture. The action of the Satake category by Hecke functors at $x\in X$ preserves nilpotent sheaves and is locally constant as $x$ varies in $X$. This compatibility is required for the Betti Langlands equivalence to identify Hecke functors with Wilson-line operators; the source expects a proof but does not provide one.