Birational torus conjecture for symplectic reduction of a conjugation and dressing orbit

Let $G$ be a semisimple group, let $B_+$ and $B_-$ be opposite Borel subgroups, let ${\mathcal O}\subset G$ be a generic conjugation $G$-orbit, and let ${\mathcal O}_-\subset B_-$ be a generic dressing orbit of $B_+$ for the standard Poisson structure. The corresponding symplectic reduction is

$$X({\mathcal O},{\mathcal O}_-)={\mathcal O}\times{c\mathcal O}_-///B_+.$$

Birational torus conjecture. There is a birational correspondence between the variety $({\mathcal O}\times {\mathcal O}_-)///B_+$ described above and a symplectic torus of the same dimension.

This conjecture links the multiplicities of restrictions of representations to the geometry of symplectic varieties arising from Hamiltonian reduction. The source gives heuristic motivation from quantization, but no resolution is stated.