Adams conjecture for local theta correspondence

Let $m,n\in\mathbb{Z}_{\geq 0}$, let $G_n=\operatorname{Sp}_{2n}(F)$ and $H_m^\pm=\operatorname{O}_{2m}^\pm(F)$ with $m>n$, and set $\alpha=2m-2n-1$. Let $\Pi(G_n)$ and $\Pi(H_m^\pm)$ denote the sets of equivalence classes of complex irreducible admissible representations, and let $\theta_{-\alpha}^\pm$ be the local theta correspondence. For a local Arthur parameter $\psi$ of $G_n$, let $\Pi_\psi$ be its local Arthur packet. If $\chi_W$ is the trivial representation of $W_F$, $\chi_V$ is the specified quadratic character, and

$$\psi_\alpha=(\chi_W\chi_V^{-1}\otimes\psi)\oplus\chi_W\otimes S_1\otimes S_\alpha,$$

then Adams conjecture. If $\pi\in\Pi_\psi$ and $\theta_{-\alpha}^\pm(\pi)\neq 0$, then

$$\theta_{-\alpha}^\pm(\pi)\in\Pi_{\psi_\alpha}.$$

This predicts that local theta correspondence preserves local Arthur packets after replacing the original parameter by $\psi_\alpha$. The source recalls this as Adams's prediction; its resolution is not specified here.