Local constancy of automorphic categories in contractible analytic families

Let $\pi:\mathfrak{X}\to S$ be a family of curves, let $U\to S^{an}$ be an analytic étale map from a complex manifold, and let $u\in U$. Let $\pi_U:U\times_S\mathfrak{X}\to U$ be the base-changed family, let $\operatorname{Bun}_G(\pi_U)$ be its moduli stack of $G$-bundles, and let $\mathcal{N}_{\pi_U}$ be the corresponding universal nilpotent cone. For the inclusion of the fiber $i_u:\operatorname{Bun}_G(\mathfrak{X}_u)\hookrightarrow\operatorname{Bun}_G(\pi_U)$, restriction gives

$$i_u^*:\mathit{Sh}_{\mathcal{N}_{\pi_U}}(\operatorname{Bun}_G(\pi_U))\to\mathit{Sh}_{\mathcal{N}_u}(\operatorname{Bun}_G(\mathfrak{X}_u)).$$

Local constancy conjecture. If $U$ is contractible, then the restriction functor $i_u^*$ is an equivalence. This is the fiberwise formulation of local constancy for the automorphic category with global nilpotent singular support; the supplied text gives no resolution status.