The generalized Kähler Calabi–Yau existence, uniqueness, and rigidity conjecture

Let $(M^{4n}, g, I, J)$ be a nondegenerate generalized Kähler manifold, and consider its $\Omega$-Hamiltonian deformation class. Generalized Kähler Calabi–Yau conjecture. There exists a nondegenerate generalized Kähler structure $(g',I,J')$ in this deformation class solving the generalized Kähler Calabi–Yau equation. Moreover, this resulting generalized Kähler structure is unique and hyper-Kähler. This conjecture is motivated by the Calabi–Yau theorem. The source establishes uniqueness and hyper-Kähler rigidity for any solution, while existence remains open.