The main conjecture on endoscopy correspondences

Let $\tau$ be the self-dual cuspidal automorphic datum and let the global Arthur parameters $\psi_1$, $\psi_2$, and $\psi=\psi_1\boxplus\psi_2$ determine the relevant classical groups $G_0$, $H$, and $G$. Let $\sigma$ and $\pi$ be irreducible automorphic representations of $H(\mathbb A)$ and $G(\mathbb A)$ occurring in the discrete spectrum. Main conjecture on endoscopy correspondences. There should exist a family of automorphic kernel functions $\mathcal K_{\psi_1;H,G}(h,g)$ such that, whenever the displayed period is convergent and nonzero, one has

$$\int_{H(F)\backslash H(\mathbb A)}\int_{G(F)\backslash G(\mathbb A)}\mathcal K_{\psi_1;H,G}(h,g)\varphi_\sigma(h)\overline{\varphi_\pi(g)}\,dh\,dg\neq0$$

if and only if $\sigma\in\widetilde\Pi_{\psi_2}(\epsilon_{\psi_2})$ precisely when $\pi\in\widetilde\Pi_{\psi}(\epsilon_{\psi})$. This proposes a uniform automorphic-integral-transform realization of elliptic endoscopy correspondences through Arthur packets; the source gives no resolution, so the claim remains open.