Langlands correspondence for reductive groups over finite fields

Let $G$ be a reductive group over the finite field $k$, let $G^{\vee}$ be its dual group, and let $\ell$ be a prime distinct from the characteristic of $k$. Write $\operatorname{Irr}_{\overline{\mathbb{Q}}_{\ell}}(G(k))$ for the irreducible $\overline{\mathbb{Q}}_{\ell}$-representations of $G(k)$, and let $\Phi_{\overline{\mathbb{Q}}_{\ell}}(G)_{\mathrm{sp}}$ denote the equivalence classes of special Frobenius-semisimple $L$-parameters. For $\varphi\in\Phi_{\overline{\mathbb{Q}}_{\ell}}(G)_{\mathrm{sp}}$, put $A_{\varphi}=Z_{\overline{A}(\varphi_0)}(\overline{\varphi(\sigma_q)})$, with $\varphi_0=\varphi|_{\mathbb{G}_{\mathrm{a}}\times I_k}$. Langlands correspondence. There is a natural map

$$\mathcal{L}_G\colon \operatorname{Irr}_{\overline{\mathbb{Q}}_{\ell}}(G(k))\to\Phi_{\overline{\mathbb{Q}}_{\ell}}(G)_{\mathrm{sp}}$$

such that, for $\varphi\in\Phi_{\overline{\mathbb{Q}}_{\ell}}(G)_{\mathrm{sp}}$, there is a bijection between $\mathcal{L}_G^{-1}(\varphi)$ and $\operatorname{Irr}_{\overline{\mathbb{Q}}_{\ell}}(A_{\varphi})$. This formulates the expected parametrization of irreducible representations of $G(k)$ by special Langlands parameters together with irreducible representations of the associated component group; the supplied material does not establish whether the statement has been proved or remains open.