The local Langlands correspondence for disconnected reductive groups

Let $G$ be a connected, quasi-split reductive group over $F$, let $(G',\xi,z)$ be a rigid inner twist of $G$, and let $\Phi_\mathrm{e}(G;[z])$ denote the set of $\widehat{G}$-conjugacy classes of $[z]$-relevant enhanced L-parameters. Here $\operatorname{Irr}(G'(F))$ is the set of irreducible representations of $G'(F)$. The local Langlands correspondence. There exists a bijection

$$\operatorname{LLC}_{(G',\xi,z)}\colon \operatorname{Irr}(G'(F))\to \Phi_\mathrm{e}(G;[z]);\quad \pi\mapsto [\phi_\pi,\rho_{(z,\pi)}]$$

and it satisfies a lot of properties. The statement proposes a local Langlands parametrization for irreducible representations of rigid inner forms by relevant enhanced L-parameters; the parser supplies no evidence resolving it, so its status remains open.