  We study the quaternionic Calabi-Yau problem in HKT geometry introduced by
Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian
hypercomplex structure. We show that the quaternionic Monge-Amp\`ere equation
on these manifolds can always be solved for every data which is invariant by
the action of a 3-dimensional torus.
