Alesker–Verbitsky's quaternionic Calabi conjecture

Let $(M,I,J,K)$ be an HKT manifold of quaternionic dimension $n$, with HKT form $\Omega$ and operators $\partial$ and $\partial_J=J^{-1}\circ\overline{\partial}\circ J$. A q-positive section of $\Lambda^{2n,0}_{I,\mathbb{R}}(M)$ has the form $e^F\Omega^n$ for some $F\in C^\infty(M)$. Alesker–Verbitsky's quaternionic Calabi conjecture. Given any such section, there exist an HKT metric $g$ on $(M,I,J,K)$ and a function $\phi\in C^\infty(M)$ such that the associated HKT form is $\Omega+\partial\partial_J\phi$ and, for some $A>0$,

$$\left(\Omega+\partial\partial_J\phi\right)^n=Ae^F\Omega^n.$$

This is the quaternionic analogue of the Calabi conjecture for HKT manifolds, prescribing the top exterior power of the HKT form. The supplied text attributes the conjecture to Alesker and Verbitsky; its resolution status is not specified here.