Local geometric Langlands compatibility conjecture for $S$-duality

Let $G$ be a group with Langlands dual group $G^{\vee}$, let $\calT$ be a theory with a $G$-action, and let $\calT^{\vee}$ be its $S$-dual theory with a $G^{\vee}$-action. Let $\bfL_G$ and $\bfL_{G^{\vee}}$ denote the conjectural local geometric Langlands equivalences from categories with strong loop-group action to categories over the corresponding local-system categories. Local geometric Langlands compatibility conjecture. There are natural equivalences

$$\bfL_G(\calC_C(\calT))\simeq \calC_H(\calT^{\vee}/G^{\vee}),\qquad \bfL_{G^{\vee}}(\calC_C(\calT^{\vee}))\simeq \calC_H(\calT/G).$$

This proposes compatibility between $S$-duality of three-dimensional theories and local geometric Langlands duality; both the underlying local Langlands framework and these equivalences are presented as conjectural.