Containment of the open-orbit Schwartz space

Let $F$ be the field in the paper, let $P$ be the relevant parabolic subgroup, and let $X_P^{\circ}(F) \subset X_P(F)$ be the open subset appearing in the construction of the Schwartz spaces. Denote by $\mathcal{S}(X_P^{\circ}(F))$ and $\mathcal{S}(X_P(F))$ the corresponding Schwartz spaces. Containment conjecture. The Schwartz space of the open subset is contained in the Schwartz space of the full space:

$$\mathcal{S}(X_P^{\circ}(F)) \subseteq \mathcal{S}(X_P(F)).$$

This is presented as a conjecture because the inclusion is not clear from the Mellin-inversion construction. The supplied resolution evidence states that it was proved in both the nonarchimedean and archimedean cases.