Ngô's compact-support conjecture for the -Schwartz space

Assume that $\rho$ factors through $\widehat{G}(\mathbb{C})\rtimes\operatorname{Gal}(E/F)$ for a finite Galois extension $E/F$, and let $M_\rho$ be Ngô's reductive monoid attached to $\rho$, with unit group $G$ and canonical open immersion $G\to M_\rho$. Let $\mathcal{S}_\rho(G(F))$ be the Braverman–Kazhdan Schwartz space. Ngô's compact-support conjecture. For every $f\in\mathcal{S}_\rho(G(F))$, the support of $f$ is contained in a compact subset of $M_\rho(F)$. This conjecture is motivated by global applications, where compactness in the reductive monoid would control the support of local test functions; the source gives no resolution status.