Clozel's existence conjecture for Galois representations attached to algebraic automorphic representations

Let $F$ be a number field, let $n\geq 1$, and let $G_F=\operatorname{Gal}(\overline{F}/F)$. Let $\pi$ be a cuspidal, algebraic automorphic representation of $\operatorname{GL}_n(\mathbf{A}_F)$, with finite part $\pi^\infty$ and Satake parameter $t_{\pi_v}^T$ at an unramified finite place $v$. Clozel's conjecture. The finite part $\pi^\infty$ is defined over a number field $K_\pi\leq\mathbf{C}$, and for every prime $p$ and embedding $K_\pi\to\overline{\mathbf{Q}}_p$ there exists a continuous, semi-simple representation

$$r_{\pi,p}\colon G_F\to\operatorname{GL}_n(\overline{\mathbf{Q}}_p)$$

such that, for every finite place $v\nmid p$ at which $\pi_v$ is unramified, the restriction $r_{\pi,p}|_{G_{F_v}}$ is unramified and $r_{\pi,p}(\operatorname{Frob}_v)^{\operatorname{ss}}$ is conjugate to $t_{\pi_v}^T$. This is the expected global Langlands correspondence for algebraic automorphic representations of general linear groups; the existence of these representations was known in the CM case through work of Harris--Lan--Taylor--Thorne and Scholze, while the conjecture in this general formulation motivates the refined local-global compatibility statement below.