Locality conjecture for the Schwartz space on an L-monoid

Let $G$ be a reductive group over $F$, let $\rho$ be the representation defining the reductive monoid $X_\rho$, and let $C^(X_\rho(F))$ be the space of locally constant functions on $X_\rho(F)$, viewed as functions on $G(F)$ by restriction. Locality conjecture. The Schwartz space $\mathcal{S}_\rho(G(F))$ is a $C^\infty(X_\rho(F))$-module under the usual function multiplication.

This conjectural locality property would describe the compatibility of the Schwartz space with the geometry of the monoid compactification. It is not discussed or resolved in the source.