Euler–Schwartz inclusion conjecture for L-monoids

Let $G$ be a reductive group over $F$, let $\rho$ define the reductive monoid $X_\rho$, and write $\mathcal{S}_{\mathrm{ES}}(X_\rho(F)):=C^\infty_c(X_\rho(F))$. View these functions on $G(F)$ by restriction. Euler–Schwartz inclusion conjecture. The space $\mathcal{S}_{\mathrm{ES}}(X_\rho(F))$ is contained in $\mathcal{S}_\rho(G(F))$, up to a unique central twist.

The source proves that $\mathcal{S}_{\mathrm{ES}}(X_\rho(F))$ is contained in the smooth Schwartz space and expects the stated inclusion in general. The conjecture remains open.