Uniqueness of twisted linear periods over Archimedean division algebras

Let $\text{DD}$ be $\mathbb{R}$, $\text{C}$, or $\text{HH}$, and let $H={\text{GL}}_p(\text{DD})\times {\text{GL}}_q(\text{DD})$ be the block diagonal subgroup of $G={\text{GL}}_{p+q}(\text{DD})$. We prove that $\dim {\text{Hom}}_H(\pi ,\eta )\le 1$ for every irreducible Casselman–Wallach representation $\pi$ of $G$ and every smooth character $\eta$ of $H$.