Primitive decomposition of Bott-Chern and Dolbeault harmonic $(k,k)$-forms on compact almost K\"ahler manifolds

We consider the primitive decomposition of $\bar \partial, \partial$, Bott-Chern and Aeppli-harmonic $(k,k)$-forms on compact almost K\"ahler manifolds $(M,J,\omega)$. For any $D \in \{\bar\partial, \partial, BC, A\}$, we prove that the $L^k P^0$ component of $\psi \in \mathcal{H}_{D}^{k,k}$, is a constant multiple of $\omega^k$. Focusing on dimension 8, we give a full description of the spaces $\mathcal{H}_{BC}^{2,2}$ and $\mathcal{H}_{A}^{2,2}$, from which follows $\mathcal{H}^{2,2}_{BC}\subseteq\mathcal{H}^{2,2}_{\partial}$ and $\mathcal{H}^{2,2}_{A}\subseteq\mathcal{H}^{2,2}_{\bar\partial}$. We also provide an almost K\"ahler 8-dimensional example where the previous inclusions are strict and the primitive components of an harmonic form $\psi \in \mathcal{H}_{D}^{k,k}$ are not $D$-harmonic, showing that the primitive decomposition of $(k,k)$-forms in general does not descend to harmonic forms.