The C-arithmeticity conjecture for automorphic representations

Let $F$ be a number field, let $G$ be a connected reductive group over $F$, and let $cpi$ be an automorphic representation of $G$. Recall that $cpi$ is $C$-arithmetic if almost all of its unramified local representations are defined over a number field, and that $C$-algebraicity is the corresponding archimedean algebraicity condition.

The $C$-arithmeticity conjecture. The representation $cpi$ is $C$-arithmetic if and only if it is $C$-algebraic.

This is the $C$-side analogue of the preceding arithmeticity claim and is intended to relate fields of definition of finite-place representations to algebraicity at infinity. The paper later proves that the two conjectures are equivalent in the appropriate simultaneous formulation, but does not resolve them.