The stable-Arthur-packet conjecture for the first failure of Adams functoriality

Let $(G,H)$ be a dual pair, let $\pi$ be a representation of $G$ of Arthur type, and let $\psi^{\max}(\pi)$ be the maximal Arthur parameter attached to $\pi$. Let $\alpha_0$ be the minimum among all positive suitable integers $\alpha$ such that $\theta_{-\alpha}^-(\pi)\in\Pi_{(\psi^{\max}(\pi))_\alpha}$. The stable-Arthur-packet conjecture. If $\alpha_0\geq3$, then $\theta_{-(\alpha_0-2)}^-(\pi)$ is not of Arthur type.