Existence of descent for cuspidal automorphic representations

Let $E/K$ be a cyclic extension of global fields, and let $\rho: \Gamma_K \to \widehat{G}(\overline{\mathbb{Q}}_l)$ be a continuous, almost everywhere unramified homomorphism with Zariski-dense image. Suppose that there exists a cuspidal automorphic representation $\pi_E$ of $G(\mathbb{A}_E)$ over $\overline{\mathbb{Q}}_l$ such that, at each place of $E$ where $\pi_E$ and $\rho|_{\Gamma_E}$ are unramified, they correspond under the unramified local Langlands correspondence. Existence of descent. Then there exists a cuspidal automorphic representation $\pi_K$ of $G(\mathbb{A}_K)$ such that, at each place of $K$ where $\pi_K$ and $\rho$ are unramified, they correspond under the unramified local Langlands correspondence.

This descent statement is intended to produce global automorphic representations over $K$ from compatible automorphic data after a cyclic extension. Assuming it, the paper obtains that the semisimple local Langlands map has an image containing all $\widehat{G}$-irreducible homomorphisms; the conjecture itself is not resolved in the supplied text.