The anomaly-free symplectic representation conjecture for strongly tempered spherical varieties

Let $(G,H)$ be a strongly tempered spherical pair that is the Whittaker induction of $(G_0,H_0,\xi)$, with $G$ admitting a quasi-split pure inner form $G_{qs}$. Assume that $(\operatorname{Res}_{E/F}G_0,\operatorname{Res}_{E/F}H_0)$ is strongly tempered for every finite extension $E/F$. Let $X=H\backslash G$, let ${}^LG_X={}^LG/Z_{G,H}$, and let $\rho_X:{}^LG_X\rightarrow \operatorname{GL}(V)$ be the representation associated with $(G,H,\xi)$. The representation $\rho_X$ is symplectic and anomaly free under endoscopy.

This conjectural property is used to define square roots of local epsilon factors and is motivated by the relation between spherical periods and automorphic $L$-functions. Its general validity is assumed in the paper.