Automorphic twisting conjecture for cohomological cuspidal representations

Let $G$ be the reductive group in the source, let $K$ be the chosen model of a maximal compact subgroup, and let $M$ be a rational $G$-representation. For a factorizable automorphic representation $\Pi$ and $\sigma\in\operatorname{Aut}(\mathbb{C}/\mathbb{Q}_K)$, define

$$\Pi^\sigma:=\Pi_\infty^\sigma\widehat\otimes\Pi_{(\infty)}^\sigma,$$

where $\Pi_\infty^\sigma$ is the Casselman–Wallach completion of $(\Pi_\infty^{(K)})^\sigma$. Automorphic twisting conjecture. If $\Pi$ is an irreducible cuspidal automorphic representation of $G(\mathbb{A})$ that is cohomological with respect to $M$, then $\Pi^\sigma$ is cuspidal automorphic for every $\sigma\in\operatorname{Aut}(\mathbb{C}/\mathbb{Q}_K)$. This predicts preservation of cuspidal automorphy under the indicated Galois twisting operation; the source cites evidence for broader classes in groups of Hermitian type, while the stated conjecture remains unresolved there.