Alesker–Verbitsky conjecture for the quaternionic Monge–Ampère equation

Let $(M,I,J,K,g)$ be a compact, connected HKT manifold, and let $n$ be its quaternionic dimension. Write the quaternionic Monge–Ampère equation as

$$(\Omega+\partial\partial_J\phi)^n=e^f\Omega^n,\qquad \Omega+\partial\partial_J\phi>0.$$

Suppose that there exists a nonvanishing $I$-holomorphic $(2n,0)$-form $\Theta$ on $M$. For a real smooth function $f$, impose the necessary condition

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

Alesker–Verbitsky conjecture. For every such $f$, the quaternionic Monge–Ampère equation admits a unique smooth solution $\phi$, up to addition of a constant.

This is the quaternionic analogue of the Calabi–Yau theorem for HKT manifolds. The source presents it as an open conjecture, while proving the result on compact hyperKähler manifolds; the general HKT case remains unresolved.