The direct-sum convolution conjecture for local L-factor Schwartz spaces

Let $G$ be a reductive group over the local field $\sF$, let $\rho_1$ and $\rho_2$ be representations of the dual group, and let ${\mathcal{S}}_{\rho_i}(G)$ denote the corresponding Schwartz spaces. Let $\star$ denote convolution of functions.

Direct-sum convolution conjecture. If

$$\rho=\rho_1\oplus\rho_2,$$

then ${\mathcal{S}}_{\rho}(G)$ consists of functions of the form $\phi_1\star\phi_2$ with $\phi_i\in{\mathcal{S}}_{\rho_i}(G)$.

This conjecture proposes compatibility of the Schwartz-space construction with direct sums of representations. No resolution is supplied in the given text.