The Adams conjecture for local theta correspondence and Arthur packets

Let $G$ and $H$ be the members of a reductive dual pair, let $\theta_{-\alpha}(pi)$ denote the theta lift of an irreducible representation $pi$ of $G$, and let $\psi$ be an Arthur parameter such that $pi$ is contained in the Arthur packet attached to $\psi$. Let $\chi_W$ and $\chi_V$ be the characters associated with the two spaces, and let $S_1$ and $S_\alpha$ denote the corresponding irreducible representations of $\operatorname{SL}_2(\mathbb{C})$. Suppose that $\alpha>0$ is odd and that the big theta lift $\Theta_{-\alpha}(pi)$ is nonzero.

Adams's conjecture. The representation $\theta_{-\alpha}(pi)$ is contained in the A-packet parametrized by

$$\psi_\alpha=(\chi_Wchi_V^{-1}otimespsi)opluschi_Wotimes S_1\otimes S_\alpha.$$

This conjecture predicts that local theta correspondence respects Arthur's parametrization of representations. The paper studies this conjecture and its relation to the structure of local Arthur packets; no resolution is supplied here.