Surjectivity conjecture for the hyperkähler Kirwan map

Let $M$ be a hyperkähler manifold with a hyperhamiltonian $G$-action. The hyperkähler Kirwan map is the natural map

$$\kappa_{\mathrm{HK}}: H_G^\ast(M) \to H^\ast(M /\!/\!/ G).$$

Hyperkähler Kirwan surjectivity conjecture. The hyperkähler Kirwan map is surjective.

Kirwan surjectivity is well known for a large class of symplectic quotients, whereas the corresponding hyperkähler statement is less understood. The conjecture concerns whether every cohomology class of the hyperkähler quotient lifts from equivariant cohomology.