Natural embedding for distinguished representations and their derivatives

Let $G_n$ be the relevant general linear group, let $G_n'$ be its distinguished subgroup, and let $pi$ be an irreducible $G_n'$-distinguished representation of $G_n$ with depth $d$. Write $pi^{-}$ for its associated representation of $G_{n-d}$, and let $G_{n-d}'$ be the corresponding distinguished subgroup. Natural embedding conjecture. There is a natural embedding

$$\operatorname{Hom}_{G_n'}(\pi,\mathbb{C})\hookrightarrow \operatorname{Hom}_{G_{n-d}'}(\pi^{-},\mathbb{C}).$$

In particular, if $pi$ is distinguished, then $pi^{-}$ is also distinguished. The statement concerns the inheritance of distinction under passage to the representation of lower rank; the supplied text does not establish its resolution.