Local-global compatibility for automorphic Galois representations

Let $F$ be a number field, let $\pi=\otimes'_v\pi_v$ be an algebraic cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$, let $l$ be a rational prime, and fix an isomorphism $\iota:\overline{\mathbb{Q}}_l\xrightarrow{\sim}\mathbb{C}$. For each finite place $v$ of $F$, write $\operatorname{rec}_{F_v}$ for the local Langlands correspondence and, conjecturally, let $r_\iota(\pi):G_F\to\operatorname{GL}_n(\overline{\mathbb{Q}}_l)$ be the continuous semisimple Galois representation attached to $\pi$. For $v\nmid l$, its restriction gives the Frobenius-semisimplified Weil--Deligne representation $\operatorname{WD}(r_\iota(\pi)|_{G_{F_v}})^{\operatorname{F-ss}}\otimes_\iota\mathbb{C}$. Local-global compatibility conjecture. For all finite places $v$ of $F$, we have

$$\operatorname{WD}(r_\iota(\pi)|_{G_{F_v}})^{\operatorname{F-ss}}\otimes_\iota\mathbb{C}\cong\operatorname{rec}_{F_v}(\pi_v\otimes|\det|^{\frac{1-n}{2}}).$$

This predicts that the local Weil--Deligne representation attached to the automorphic representation agrees with the one obtained from its global Galois representation. The paper proves a new ordinary rank-two case over CM fields, while the statement in this generality is not established.