Hyperkähler rotation conjecture for affine Toda moduli spaces

Let $G$ be a semisimple group over $\a9\a9$ and let $G^{\mathrm{sc}}$ be its simply-connected cover. Let $\mathcal{M}^{\circ}_G$ denote the neutral component of the affine Toda moduli space, and let $J^G_{G^{\mathrm{sc}}}$ denote the corresponding regular centralizer group scheme. Hyperkähler rotation conjecture. There exists a hyperKähler structure on $\mathcal{M}^{\circ}_G$ such that $J^G_{G^{\mathrm{sc}}}$, as a complex manifold, is obtained from $\mathcal{M}^{\circ}_G$ by hyperKähler rotation. In particular, there is a homeomorphism

$$\mathcal{M}^{\circ}_G\simeq J^G_{G^{\mathrm{sc}}}$$

as $C^\infty$-manifolds.

This is the proposed non-abelian Hodge correspondence between the Dolbeault-type affine Toda moduli space and its Betti-type regular centralizer counterpart. The source does not state a resolution of the conjecture.