Lafforgue's rational-structure conjecture for the excursion algebra action

Let $X$ be a curve with compactification $\ol{X}$, let $\sG$ be a reductive group, and let $G$ be its Langlands dual. Let

$$\on{Autom}:=\on{Funct}_c(\Bun_G^{\on{level}}(\BF_q),\sfe),$$

where $\Bun_G^{\on{level}}$ is the moduli space of $G$-bundles on $\ol{X}$ with level structure on $\ol{X}-X$. The excursion algebra $\on{Exc}(X,\sG)$ acts on $\on{Autom}$. Lafforgue's rational-structure conjecture. This action is compatible with the rational structures. This conjecture concerns the compatibility between the excursion-algebra action and rational models, extending the expected arithmetic structure in the Langlands program. It has been established using motivic methods.