Let \((M,g,J)\) be an almost Hermitian manifold. It is called an almost Kähler manifold if the Kähler form \(\Omega\) defined by \(\Omega(X,Y)= g(X,JY)\) is closed. S. I. Goldberg conjectured that a compact almost Kähler Einstein manifold is Kählerian (i.e., \(\nabla J=0)\). Although recently some progress been made, this conjecture remains open for the general case. This paper contributes to this problem. The main result is as follows: A four-dimensional compact almost Kähler manifold of constant holomorphic sectional curvature such that its curvature tensor \(R\) satisfies \(R(X,Y,Z,W)= R(JX,JY,JZ,JW)\) is Kählerian. The proof is based on a result of \textit{K. Sekigawa} and \textit{L. Vanhecke} [Ann. Mat. Pura Appl., IV. Ser. 157, 149-160 (1990; Zbl 0731.53026)] whose proof contained a gap. This has been repaired in [\textit{T. Oguro} and \textit{K. Sekigawa}, `Four-dimensional almost Kähler Einstein and *-Einstein manifolds', Geom. Dedicata (to appear)].