  We solve the quaternionic Monge-Amp\`ere equation on hyperK\"ahler manifolds.
In this way we prove the ansatz for the conjecture raised by Alesker and
Verbitsky claiming that this equation should be solvable on any hyperK\"ahler
with torsion manifold, at least when the canonical bundle is trivial
holomorphically. The novelty in our approach is that we do not assume any
flatness of the underlying hypercomplex structure which was the case in all the
approaches for the higher order a priori estimates so far. The resulting
Calabi-Yau type theorem for HKT metrics is discussed.
