  We study an odd-dimensional analogue of the Goldberg conjecture for compact
Einstein almost K\"ahler manifolds. We give an explicit non-compact example of
an Einstein almost cok\"ahler manifold that is not cok\"ahler. We prove that
compact Einstein almost cok\"ahler manifolds with non-negative $*$-scalar
curvature are cok\"ahler (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 cok\"ahler. We prove similar bounds for almost K\"ahler
Einstein manifolds that are not K\"ahler.
