Elliott–Pestun's multiplicative geometric Langlands conjecture

Let $G$ be a semisimple simply-laced complex group, let $\theta$ be a diagram automorphism of $G$, and let $C$ be a Calabi–Yau Riemann surface. For any $\tau\to\operatorname{Aut}(C)$, write $\operatorname{Diff}_{\tau^{-1}}(\operatorname{mHiggs}_{G,\theta}(C))$ for the category associated with the $\tau^{-1}$-twisted multiplicative Higgs moduli space, and $\operatorname{Diff}_{\operatorname{id}}(\operatorname{Conn}^{\tau}_{(G^\theta)^\vee}(C))$ for the corresponding category of twisted connections for the Langlands dual group $(G^\theta)^\vee$. Elliott–Pestun's multiplicative geometric Langlands conjecture. There is a derived equivalence

$$\mathbf{mGLC}:\operatorname{Diff}_{\tau^{-1}}(\operatorname{mHiggs}_{G,\theta}(C))\longrightarrow\operatorname{Diff}_{\operatorname{id}}(\operatorname{Conn}^{\tau}_{(G^\theta)^\vee}(C)).$$

This conjecture is the multiplicative, diagram-automorphism-twisted form of geometric Langlands duality, predicting that S-duality exchanges $G^\theta$ with its Langlands dual and $\tau$ with $\tau^{-1}$. The supplied source gives no resolution, so the conjecture remains open.