Kim's conjecture on local constituents of residual Arthur packets

Let $G$ be a group of type $E_n$, let $G^{\vee}$ be its dual group, and let $\psi:SL_2(\mathbb{C})\to G^{\vee}$ be an Arthur parameter such that its global Arthur packet $\Pi_\psi$ contains a residual representation of $G$. Let $\mathcal{O}$ be the nilpotent orbit of $G^{\vee}$ associated with $\psi$. For a place $v$, let $\Pi_{\psi_v}^{\mathrm{res}}\subseteq\Pi_{\psi_v}$ consist of the representations occurring as local constituents of irreducible residual representations in the residual spectrum attached to the trivial character of $T(\mathbb{A})/T(F)$, and let $\operatorname{Springer}(\mathcal{O})$ denote the image of the Springer correspondence.

Kim's conjecture. The parameter $\psi$ is associated with a distinguished orbit $\mathcal{O}$ of $G^{\vee}$, and

$$\Pi_{\psi_v}^{\mathrm{res}}=\operatorname{image}\bigl(\operatorname{Springer}(\mathcal{O})\bigr)$$

under the local Arthur correspondence.

This conjecture predicts which members of the local Arthur packet occur in residual representations attached to the trivial character. The supplied text gives no resolution status.