Primitive automorphic base-change and descent conjecture for perfect central extensions

Let $E/F$ be a Galois extension of number fields and let $n$ be an integer such that

$$\mathrm{Gal}(E/F) \text{ is the universal perfect central extension of a finite simple nonabelian group,}$$

and, for every divisor $m\mid n$, there are no nontrivial irreducible representations $\mathrm{Gal}(E/F)\to\mathrm{GL}_m(\mathbb{C})$. Let $E\geq F'\geq F$ be a subfield. An automorphic representation is $E$-primitive in the sense of the paper. Primitive base-change and descent conjecture. Every $E$-primitive automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$ admits unique base changes $\pi_{F'}$ to $\mathrm{GL}_n(\mathbb{A}_{F'})$ and to $\mathrm{GL}_n(\mathbb{A}_E)$, with $\pi_{F'}$ $E$-primitive, and base change induces an injection

$$b_{E/F'}:\Pi_n^{\mathrm{prim}}(E/F)\longrightarrow\Pi_n^{\mathrm{prim}}(E/F'),\qquad \pi\longmapsto\pi_{F'}.$$

Moreover, if $\pi'$ is cuspidal on $\mathrm{GL}_n(\mathbb{A}_{F'})$, its base change $\pi'_E$ is cuspidal and $\mathrm{Gal}(E/F)$-invariant, then $\pi'_E$ descends to an automorphic representation of $\mathrm{GL}_n(\mathbb{A}_F)$. This is presented as an elaboration of Langlands functoriality; no resolution is supplied in the source.