Solvable base-change and descent conjecture

Let $E/F'$ be a solvable extension of number fields, and let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ that is $\mathrm{Gal}(E/F')$-invariant. Let a character of $E^{\times}\backslash\mathbb{A}_E^{\times}$ be viewed as a Hecke character, and let $\mathrm{Gal}(E/F')^{\wedge}$ denote the character group. Solvable base-change and descent conjecture. There is a $\mathrm{Gal}(E/F')$-invariant character

$$\chi\in (E^{\times}\backslash\mathbb{A}_E^{\times})^{\wedge}$$

such that $\Pi\otimes\chi$ descends to $\mathrm{GL}_n(\mathbb{A}_{F'})$. If $H^2(\mathrm{Gal}(E/F'),\mathbb{C}^{\times})=0$, $\chi$ may be chosen trivial. Conversely, if $\pi'_1$ and $\pi'_2$ are cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{A}_{F'})$ that both base change to a cuspidal representation $\Pi$ of $\mathrm{GL}_n(\mathbb{A}_E)$, then there is a unique $\chi\in\mathrm{Gal}(E/F')^{\wedge}$ such that $\pi'_1\cong\pi'_2\otimes\chi$. The statement is known in the special cases mentioned immediately before it in the source, but is open in general.