Main conjecture for cuspidal automorphic modules

Let $G_n$ be a classical group over $F$ and a pure inner $F$-form of an $F$-quasisplit classical group $G_n^*$. For $\pi\in{\mathcal {A}}_{\mathrm{cusp}}(G_n)$ in the global Arthur packet $\widetilde{\Pi}_\phi(G_n)$ attached to a $G_n$-relevant, generic global Arthur parameter $\phi\in\widetilde{\Phi}_2(G_n^*)$, there should exist a classical group $H_m$ and $\sigma\in{\mathcal {A}}_{\mathrm{cusp}}(H_m)$ satisfying the relevance and pure-inner-form conditions in the source, with $\sigma$ in a global Arthur packet attached to an $H_m$-relevant generic parameter $\phi'\in\widetilde{\Phi}_2(H_m^*)$, such that

$$\pi\cong{\mathcal D}^{{\mathcal {O}}_{\kappa_0}}_n(\tau;\sigma),$$

where $G_n=G_n^{{\mathcal {O}}_{\kappa_0}}$ and $\tau=\tau_1\boxplus\cdots\boxplus\tau_r$ is the irreducible isobaric representation associated to $\phi$. This asserts that every relevant generic cuspidal automorphic representation is obtained by the proposed twisted automorphic descent construction. The excerpt presents this as the main conjecture and gives no resolution.