On second non-HLC degree of closed symplecitc manifold

In this note, we show that for a closed almost-K\"{a}hler manifold $(X,J)$ with the almost complex structure $J$ satisfies $\dim\ker P_{J}=b_{2}-1$ space of de Rham harmonic forms is contained in symplectic-Bott-Chern forms. particular, suppose $X$ four-dimension, if self-dual Betti number $b^{+}_{2}=1$, then prove second non-HLC degree measures gap between and