“…Animated by the study of "canonical" HKT metrics, in analogy to the Calabi conjecture [15] proved by Yau in [56], Alesker and Verbitsky proposed in [6] to study the quaternionic Monge-Ampère equation: on a compact HKT manifold, where H ∈ C ∞ (M, R) is given, while (ϕ, b) ∈ C ∞ (M, R) × R + is the unknown. Even if the solvability of the quaternionic Monge-Ampère equation is still an open problem in its general form, several partial results are available in the literature [2,3,4,6,10,19,22,23,46,58]. A nice geometric application of the solvability of equation ( 1) is the existence of a unique balanced metric g on a compact HKT manifold (M, I, J, K, g) with holomorphically trivial canonical bundle with respect to I such that the form Ω induced by g belongs to the class {Ω + ∂∂ J ϕ} (see [54]).…”