Arthur's existence and uniqueness conjecture for distinguished unipotent parameters

Let $G$ be the reductive group under consideration, let $X$ be the underlying curve, and let $\check{G}$ be its Langlands dual group. Let

$$\sigma:\operatorname{SL}_2\to\check{G}$$

be a distinguished unipotent Arthur parameter, meaning that its image is not contained in any proper Levi subgroup of $\check{G}$. For a representation $V$ of $\check{G}$ and a point $x\in X$, write $h=\operatorname{Sat}_x(V)$, and let $V_{\mathbb G_m}$ denote $V$ with the grading induced by $\mathbb G_m\subset\operatorname{SL}_2\to\check{G}$. Arthur's conjecture. There exists an essentially unique derived sheaf $\mathcal F$ on $\operatorname{Bun}_G$ such that

$$h*\mathcal F=V_{\mathbb G_m}\otimes\mathcal F$$

for every $x\in X$ and every representation $V$ of $\check{G}$. Arthur's conjecture predicts the geometric analogue of the existence and uniqueness of unipotent automorphic representations; the supplied text gives no resolution of this distinguished-parameter case.