Local trace conjecture for restricted loop-group categories

Let $G$ be a reductive group and let $\fL(G)\operatorname{Cat}_{\operatorname{restr}}$ be the restricted 2-category of categories acted on by its loop group. Let $\operatorname{Frob}$ denote Frobenius, and let $\operatorname{Tr}_{\operatorname{DGCat}}(\operatorname{Frob},\fL(G)\operatorname{Cat}_{\operatorname{restr}})$ be the corresponding categorical trace. Local trace conjecture. The natural map

$$\operatorname{emb.restr}\bigl(\operatorname{Tr}_{\operatorname{DGCat}}(\operatorname{Frob},\fL(G)\operatorname{Cat}_{\operatorname{restr}})\bigr)\to \operatorname{Tr}_{\operatorname{AGCat}}(\operatorname{Frob},\fL(G))$$

is an isomorphism. This conjecture compares the restricted and unrestricted local trace constructions; the source gives no resolution evidence.