Rationality conjecture for canonical local periods

Let $E$ be a field, let $G$ be a reductive group over a local field $F$, and let $Y=H\backslash G$ be a spherical variety. For a smooth admissible $G(F)$-representation $\pi$ over $E$, let $\mathcal{E}(\pi)$ be the set of embeddings $\tau:E\to\mathbb{C}$ such that $\pi_\tau$ is irreducible and occurs in the Plancherel decomposition of $L^2(Y(F))$. Define

$$(\pi\otimes_E\pi^\vee)^\star=\left\{\sum_i v_i\otimes v^i\in\pi\otimes\pi^\vee\mid\sum_i v^i(v_i)\neq0\right\}$$

and let $Q(\pi\otimes_E\pi^\vee)$ be the corresponding localization. Rationality conjecture for canonical local periods. If $\mathcal{E}(\pi)$ is nonempty, there is a unique $E$-linear map

$$Q_\pi:Q(\pi\otimes_E\pi^\vee)\longrightarrow E$$

whose scalar extension under every $\tau\in\mathcal{E}(\pi)$ is $Q_{\pi_\tau}$. This asserts that the canonical local period is defined over $E$ whenever at least one complex realization occurs in the Plancherel decomposition; the source gives no resolution status.