Topological invariance of the Betti automorphic category

Let $G$ be a complex reductive group and let $X$ be a smooth complex projective curve. Let $\mathcal Shv_{\mathcal N}(Bun_G(X))$ be the Betti category of nilpotent sheaves. Topological invariance of the Betti automorphic category. The category $\mathcal Shv_{\mathcal N}(Bun_G(X))$ depends on $X$ only through its underlying topological surface. The conjecture would account for the topological nature of the spectral character stack and is stated to be nontrivial because the geometry of the global nilpotent cone can vary with the algebraic curve.