Nonsplit Langlands conjecture for $n=4$

Let $G_4=\operatorname{Res}_{\mathbf C[[t^{1/4}]]/\mathbf C[[t]]}\mathrm{PGL}_2$. Let $\overline{\mathcal O}$ be the minimal nilpotent coadjoint orbit closure of $\mathfrak{so}_8$, equipped with the stated $\Sigma_4$-action. The action of $\mathrm{SL}_2^{\times4}\subseteq\mathrm{SO}_8$ descends to an action of $\mathrm{SL}_2$ on $\overline{\mathcal O}/\!/\Sigma_4$. Nonsplit Langlands conjecture for $n=4$. For this action and a certain grading on $\overline{\mathcal O}/\!/\Sigma_4$, there is a fully faithful functor

$$\operatorname{Perf}^{\mathrm{sh}}\!\left((\overline{\mathcal O}/\!/\Sigma_4)/\mathrm{SL}_2(-2\rho)\right)\hookrightarrow\operatorname{Shv}^{c}_{G_4[[t]]}\!\left(G_4((t))/\mathrm{PGL}_2((t));\mathbf Q\right).$$

This is the $n=4$ specialization of the nonsplit Langlands expectation and is not established in the source.