The locality conjecture for the local L-factor Schwartz space

Let $G$ be a reductive group over the local field $\sF$, let $\overline{G}$ be the relevant compactification, and let ${\mathcal{S}}_{\rho}(G(\sF))$ be the proposed Schwartz space. A sheaf on $\overline{G}(\sF)$ is understood to carry the indicated $G(\sF)\times G(\sF)$-equivariance.

Locality conjecture. There exists a $G(\sF)\times G(\sF)$-equivariant sheaf ${\mathbf S}_{\rho}$ of complex vector spaces on $\overline{G}(\sF)$ such that ${\mathbf S}_{\rho}|_{G(\sF)}$ is the sheaf of locally constant functions on $G(\sF)$ and

$${\mathcal{S}}_{\rho}(G(\sF))=\Gamma_c(\overline{G}(\sF),{\mathbf S}_{\rho}).$$

This asserts that the proposed test-function space is local on the compactification and can be recovered as compactly supported global sections of a canonical equivariant sheaf. The given text does not provide a resolution.