Compactly supported functions conjecture for Schwartz spaces on L-monoids

Let $G$ be a reductive group over $F$, let $\rho$ define the reductive monoid $X_\rho$, and let $C^\infty_c(X_\rho(F))$ be the space of compactly supported locally constant functions on $X_\rho(F)$, viewed as functions on $G(F)$ by restriction. Compact-support inclusion conjecture. The space $C^\infty_c(X_\rho(F))$ is contained in $\mathcal{S}_\rho(G(F))$, up to a unique central twist.

This is proposed as an intermediate problem for understanding the $G(F)\times G(F)$-module structure of $\mathcal{S}_\rho(G(F))$. The source does not provide a resolution.