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For a compact K\"ahler manifold, it is well-established that its de Rhamcohomology satisfies the Hard Lefschetz condition, which is reflected in theequality between the Betti numbers and the Hodge numbers. A special subclass ofsymplectic manifolds also adheres to this condition. Cirici and Wilson\cite{CW20} employ the variant Hodge number to propose a sufficient criterionfor compact almost K\"ahler manifolds to satisfy this condition. In this paper,we show that this condition is only sufficient by presenting examples ofcompact almost K\"ahler manifolds that fulfill the Hard Lefschetz conditionwhile violating the equality between the variant Hodge numbers and Bettinumbers, that is, \[b^1>2h^{1,0}.\] This phenomenon contrasts with the behaviorobserved in compact K\"ahler manifolds.

Compatible almost complex structures on the Hard Lefschetz condition