Arthur functoriality conjecture for Levi subgroups

Let $G$ be a semisimple algebraic group over ${\mathbb Q}$, let $L$ be the Levi subgroup associated with a $\theta$-stable parabolic subalgebra $\mathfrak{q}$, let $\tau$ be a unitary representation of $L$, and let $\pi$ be the representation of $G$ obtained from $\tau$ by the functoriality induced by the $L$-homomorphism $\xi_L$. Write $\widehat{G}_{\rm Aut}$ and $\widehat{L}_{\rm Aut}$ for the automorphic duals. Arthur functoriality conjecture. The representation $\pi$ belongs to $\widehat{G}_{\rm Aut}$ if and only if, after conjugating $L$ in $G$, the inclusion $L\subset G$ is defined over ${\mathbb Q}$ and $\tau\in\widehat{L}_{\rm Aut}$. The claim expresses the expected automorphic compatibility of Arthur functoriality with rational Levi inclusions and is used in the source to deduce automorphy of cohomological representations; it remains open there.