The quaternionic Calabi conjecture for compact HKT-manifolds

Let $(M^{4n},I,J,K)$ be a compact hypercomplex manifold of real dimension $4n$, let $\Omega$ be an HKT-form, and let $f$ be a real-valued $C^\infty$ function on $M$. Assume that $(M,I)$ admits a holomorphic, non-vanishing $(2n,0)$-form $\Theta$. For an unknown real-valued function $\varphi$, consider the quaternionic Monge–Ampère equation

$$(\Omega+\partial\partial_J\varphi)^n=e^f\Omega^n.$$

Quaternionic Calabi conjecture. If the necessary condition

$$\int_M(e^f-1)\Omega^n\wedge\overline{\Theta}=0$$

is satisfied, then the equation has a $C^\infty$ solution $\varphi$. This is the quaternionic analogue of the Calabi problem for Kähler manifolds; the condition arises by integrating the equation against the holomorphic volume form, while existence of smooth solutions for general compact HKT-manifolds under this condition is the problem being conjectured.