Montonen–Olive duality conjecture for pure Yang–Mills theory

Let $G$ be a gauge group with gauge coupling $\tau$, and let $G^\vee$ be a group whose Lie algebra satisfies $\operatorname{Lie}(G^\vee)=\mathfrak{g}=\operatorname{Lie}(G)$. Let $\Lambda_w^G$ and $\Gamma_w^G$ denote the character and cocharacter lattices, and let tildes denote the dual lattices. Montonen–Olive duality conjecture. Electric-magnetic duality $S$ maps pure Yang–Mills theory with gauge group $G$ and gauge coupling $\tau$ to pure Yang–Mills theory with gauge group $G^\vee$ and gauge coupling $-1/\tau$, where

$$\Lambda_w^{G^\vee}=\widetilde{\Gamma_w^G}$$

or, dually,

$$\Gamma_w^{G^\vee}=\widetilde{\Lambda_w^G}.$$

This conjecture proposes an electric-magnetic exchange between gauge groups determined by dual character and cocharacter lattices. The supplied text does not state whether it has been proved or disproved.