Relative distinction conjecture for complex points of real reductive groups

Let $G$ be a reductive algebraic group over $\mathbb{R}$, and let $\pi$ be an irreducible representation of $G(\mathbb{C})$. Complex-real distinction conjecture. If $\operatorname{Hom}_{G(\mathbb{R})}[\pi,\omega_G]$ is nonzero, then $\pi^\sigma=\pi^\vee$. If $G$ is quasi-split over $\mathbb{R}$, this Hom-space is nonzero if and only if the parameter of $\pi$ is obtained by base change from a parameter for $G^{\rm op}$. This is the basic real-group case of the relative local Langlands conjecture and links distinction to conjugate self-duality and base change; no resolution is supplied in the source.