The automorphic Galois group conjecture for global Langlands correspondences

Let $F$ be a number field and let $L_F$ be the conjectural automorphic Galois group, with localizations compatible with the local Langlands correspondence. Write $\Pi_{\mathrm{cusp,temp}}(\operatorname{GL}(N))$ for cuspidal tempered automorphic representations of $\operatorname{GL}(N)$, and let $\Pi_{\mathrm{aut,temp}}(G)$ denote tempered automorphic representations of a quasisplit group $G$ over $F$.

Automorphic Galois group conjecture.

(a) There is a bijection

$$\{r:L_F\longrightarrow\operatorname{GL}(N,\mathbb{C})\}\xrightarrow{\sim}\{\pi\in\Pi_{\mathrm{cusp,temp}}(\operatorname{GL}(N))\},$$

from equivalence classes of bounded, irreducible, $N$-dimensional representations of $L_F$ to cuspidal tempered automorphic representations of $\operatorname{GL}(N)$, compatible with the local Langlands correspondence at each $v$.

(b) For any quasisplit group $G$ over $F$, there is a bijection

$$\{\phi:L_F\longrightarrow{}^LG\}\xrightarrow{\sim}\{\Pi_\phi\subset\Pi_{\mathrm{aut,temp}}(G)\},$$

from equivalence classes of bounded global parameters $\Phi_{\mathrm{bdd}}(G)$ to disjoint global packets, whose union contains the set $\Phi_{\mathrm{aut,temp}}(G)$ of tempered automorphic representations of $G(\mathbb{A})$, compatible with the local Langlands correspondence at each $v$.

The conjecture proposes that $L_F$ simultaneously parameterizes cuspidal tempered representations of general linear groups and tempered global packets for all quasisplit groups. The supplied text does not state which cases are known or the conjecture's current status.