  We formulate a Calabi-Yau type conjecture in generalized K\"ahler geometry,
focusing on the case of nondegenerate Poisson structure. After defining natural
Hamiltonian deformation spaces for generalized K\"ahler structures generalizing
the notion of K\"ahler class, we conjecture unique solvability of Gualtieri's
Calabi-Yau equation within this class. We establish the uniqueness, and
moreover show that all such solutions are actually hyper-K\"ahler metrics. We
furthermore establish a GIT framework for this problem, interpreting solutions
of this equation as zeros of a moment map associated to a Hamiltonian action
and finding a Kempf-Ness functional. Lastly we indicate the naturality of
generalized K\"ahler-Ricci flow in this setting, showing that it evolves within
the given Hamiltonian deformation class, and that the Kempf-Ness functional is
monotone, so that the only possible fixed points for the flow are
hyper-K\"ahler metrics. On a hyper-K\"ahler background, we establish global
existence and weak convergence of the flow.
