Refined Langlands duality conjecture for representation spaces

Let $\Gamma$ be a finite subgroup of $SU(2)$. For an Abelian central subgroup $Z\subset G$ of a connected compact simple Lie group $G$, set $H=G/Z$, with central extension

$$0\to Z\to G\to H\to 0.$$

Let $\widetilde G$ and $\widetilde H$ be the Langlands duals, so that

$$0\to Z^\wedge\to \widetilde H\to \widetilde G\to 0.$$

For the representation spaces and finite symmetry groups defined in the source, let $s:F(\Gamma;Z)\xrightarrow{\sim}F(\Gamma;Z^\wedge)$ be the swap isomorphism. Refined Langlands duality conjecture. The representation $V_Z(\Gamma,G)$ of $F(\Gamma;Z)$ and the representation $V_{Z^\wedge}(\Gamma,\widetilde H)$ of $F(\Gamma;Z^\wedge)$ are equivalent after identifying the two groups using $s$. The refinement predicts an equivalence of the corresponding representations, not merely equality of their dimensions or of homomorphism counts. The source gives no resolution of this refined statement; it is presented as a conjectural refinement of the preceding conjecture.