Joyce's coadjoint-orbit conjecture for hyperkähler structures

Let $G$ be a compact Lie group with Lie algebra $\mathfrak g$, let $G^c$ be its complexification, and let $\Omega$ be the data defining the examples associated to $G$. Denote the resulting HP-algebra and space by $A^\Omega$ and $M^\Omega$, respectively. Coadjoint-orbit conjecture. The assumption in Example 4.3 always holds; the HP-algebra $A^\Omega$ determines a hyperkähler structure on a dense open subset of $M^\Omega$; these structures include all the structures of Kronheimer, Biquard, and Kovalev as special cases, while generically they are new and do not coincide with those structures. For some $\Omega$, $M^\Omega$ is a cone in $\mathfrak g^*\otimes\mathbb I$ and $A^\Omega$ is an SGH-algebra, so $M^\Omega$ is a hyperkähler cone; these are the nilpotent orbits of Kronheimer. For every $\Omega$ there is a $\widetilde\Omega$ such that $M^{\widetilde\Omega}$ is a nilpotent orbit, the associated graded H-algebra of $A^\Omega$ is $A^{\widetilde\Omega}$, and the singular hyperkähler manifold $M^\Omega$ has an AC end with asymptotic cone $M^{\widetilde\Omega}$. The conjecture proposes an algebraic description of the coadjoint-orbit hyperkähler metrics constructed by Kronheimer, Biquard, and Kovalev; the supplied context says it was proved for Kronheimer's metrics but remained unproved for the metrics of Biquard and Kovalev.