Folding compatibility for classical multiplicative Langlands duality

Let $G$ and $G^\vee$ be Langlands dual simple Lie groups, and suppose that $G$ arose by folding the Dynkin diagram of a self-dual group $\widetilde G$. Let $\mathrm{Out}(\widetilde G)$ act simultaneously on $\widetilde G$ and on the circle $S^1_B$, under the identification

$$\operatorname{mHiggs}_{\widetilde G}(E)=\operatorname{Bun}_{\widetilde G}(E\times S^1_B).$$

Folding conjecture for classical multiplicative Langlands duality. There is an equivalence of categories

$$\operatorname{Coh}(\operatorname{mHiggs}_{G^\vee}(E)) \cong \operatorname{Coh}(\operatorname{mHiggs}_{\widetilde G}(E)^{\mathrm{Out}(\widetilde G)}),$$

compatible with the Fourier–Mukai transform relating the categories for the maximal tori. This proposes that the non-simply-laced correspondence is obtained by folding a simply-laced self-dual theory; no resolution is given.