Birational torus conjecture for the triple-orbit symplectic reduction

Let $G$ be a semisimple group, and let ${\mathcal O}_1,{c\mathcal O}_2,{c\mathcal O}_3\subset G$ be three conjugation $G$-orbits. Let $\bar{{\mathcal O}}_3$ denote the opposite symplectic variety to ${\mathcal O}_3$. The Hamiltonian reduction is

$$X({\mathcal O}_1,{c\mathcal O}_2,{c\mathcal O}_3)=({\mathcal O}_1\times{\mathcal O}_2\times\bar{{\mathcal O}}_3)///G.$$

Birational torus conjecture. There is a birational equivalence between the symplectic variety $({\mathcal O}_1\times{\mathcal O}_2\times\bar{{\mathcal O}}_3)///G$ and a complex torus of the same dimension.

This conjecture concerns the geometry underlying multiplicities of generic irreducible modules in tensor products of representations. The source motivates it through quasi-Hamiltonian reduction and quantization, but does not state a resolution.