The local constancy conjecture for automorphic categories in a family of curves

Let $M$ be a real manifold, let $\pi_M:\mathfrak{X}_M\to M$ be the induced family of curves, and let $s\in M$. Write

$$\mathit{Sh}_{\mathcal{N}}(\operatorname{Bun}_{G}(\pi_M)):=\mathit{Sh}_{\widetilde{\mathcal{N}}_{\Pi_M}}(\operatorname{Bun}_{G}(\pi_M)),$$

where $\widetilde{\mathcal{N}}_{\Pi_M}$ is the universal nilpotent cone, and let

$$i_s^*:\mathit{Sh}_{\mathcal{N}}(\operatorname{Bun}_{G}(\pi_M))\longrightarrow\mathit{Sh}_{\mathcal{N}}(\operatorname{Bun}_{G}(\mathfrak{X}_s))$$

be the restriction functor. The local constancy conjecture. If $M$ is a contractible real manifold, then the restriction functor $i_s^*$ is an equivalence. Intuitively, the conjecture says that the categories of nilpotent sheaves on the automorphic categories of the fibers vary locally constantly with $s$.