Hyperkähler structure conjecture for the cotangent bundle of higher Teichmüller space

The cotangent bundle T^*\bm\hat{\mathcal{T}}^n is obtained by Hamiltonian reduction from the hyperkähler punctual Hilbert scheme, with its given symplectic structure near the zero-section. Hyperkähler structure conjecture. The cotangent bundle T^*\bm\hat{\mathcal{T}}^n admits a hyperkähler structure near the zero-section. This is motivated by the hyperkähler structure on $\operatorname{Hilb}^n(\mathbb{C}^2)$ and the Hamiltonian-reduction description of the cotangent bundle; the conjectural inheritance is asserted only near the zero-section.