Langlands duality conjecture for homomorphisms from finite subgroups of $SU(2)$

Let $\Gamma$ be a finite group, and let $G$ be a connected compact simple Lie group. Write $N(\Gamma,G)$ for the number of homomorphisms from $\Gamma$ to $G$ up to conjugation by $G$. A connected compact simple Lie group $\widetilde G$ is Langlands dual to $G$ when its root lattice, weight lattice, and Cartan-torus character lattice are respectively the duals of the weight lattice, root lattice, and character lattice of $G$. Langlands duality conjecture. If $\Gamma$ is a finite subgroup of $SU(2)$ and $(G,\widetilde G)$ is a Langlands dual pair, then

$$N(\Gamma,G)=N(\Gamma,\widetilde G).$$

The conjecture predicts equality of the numbers of conjugacy classes of homomorphisms into Langlands-dual groups, a phenomenon motivated by four-dimensional supersymmetric quantum field theory. The source presents it as a conjecture and mentions that a more refined version and proofs of some concrete cases are available, while asking for a uniform proof in all cases.