Local constancy of the automorphic category for families of curves

Let $\pi:\mathfrak{X}\to S$ be the family of curves in the paper, let $\operatorname{Bun}_G(\pi_U)$ be the moduli stack of $G$-bundles on the base-changed family over a contractible analytic open subset $U\subset S^{an}$, and let $s\in U$. Let $\mathcal{N}_{\pi}$ and $\mathcal{N}_{\mathfrak{X}_s}$ denote the universal and fiberwise global nilpotent cones, respectively. Local constancy conjecture. Restriction along $i_s:\{s\}\hookrightarrow U$ induces an equivalence

$$\mathit{Sh}_{\mathcal{N}_{\pi}}(\operatorname{Bun}_G(\pi_U))\stackrel{\sim}{\longrightarrow}\mathit{Sh}_{\mathcal{N}_{\mathfrak{X}_s}}(\operatorname{Bun}_G(\mathfrak{X}_s)).$$

This asserts that the automorphic categories of fibers vary locally constantly in a contractible analytic family when equipped with the universal nilpotent singular-support condition. The supplied text gives no resolution status beyond presenting the statement as local constancy, so it remains open here.