Einstein almost cokahler manifolds

We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kahler manifolds. We give an explicit non-compact example of an Einstein almost cokahler manifold that is not cokahler. We prove that compact Einstein almost cokahler manifolds with non-negative $*$-scalar curvature are cokahler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cokahler. We prove similar bounds for almost Kahler Einstein manifolds that are not Kahler.