Restricted local geometric Langlands conjecture

Let $G$ be a reductive group with Langlands dual $cG$, let $\fL(G)$ be its loop group, and let $\fL(G)\operatorname{Cat}_{\operatorname{restr}}$ be the restricted 2-category of categories acted on by $\fL(G)$. Let $\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD)$ be the restricted moduli stack of local systems on a formal punctured disc, and let $\operatorname{tIndCoh}_\Nilp$ denote the corresponding 2-category with nilpotent singular support. Restricted local geometric Langlands conjecture. There exists an equivalence of 2-categories

$$\fL(G)\operatorname{Cat}_{\operatorname{restr}}\simeq \operatorname{tIndCoh}_\Nilp(\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD)).$$

This is the local categorical Langlands assertion in the restricted setting; the source gives no resolution evidence.