Cohomology ring of a closed manifold $M^{2n}$ is said to satisfy **hard Lefschetz property** if there exists an element $a\in H^2(M,\mathbb R)$ such that multiplication by $a^k$ yields an isomorphism
$$H^{n-k}(M,\mathbb R)\to H^{n+k}(M,\mathbb R)$$
for all $k=1,\dots,n$.

By the hard Lefschetz theorem cohomology ring of any compact Kahler manifold $(M,\omega)$ has this property with $a=[\omega]$.

My question is: What are the examples of (necessarily non-Kahler) Moishezon (or Fujiki class C) manifolds with cohomology ring not satisfying hard Lefschetz property?