It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H_{BC}^{\bullet,\bullet}:=\frac{\ker \partial\cap\ker\bar\partial}{\text{im }\partial\bar\partial}$ is isomorphic to the Dolbeault cohomology $H_{\bar\partial}^{\bullet,\bullet}(X)=\frac{\ker\bar\partial}{\text{im }\bar\partial}$, and the Frölicher spectral sequence of $X$ degenerates at $E_1$, then $$ H^k(X,\mathbb C)=\bigoplus_{p+q=k}H^{p,q}_{\bar\partial}(X)=\bigoplus_{p+q=k}H^{p,q}_{BC}(X), $$ so $H^{p,q}_{\bar\partial}(X)$ can be treated as a subspace of $H^k(X,\mathbb C)$ through the isomorphism between Bott-Chern and Dolbeault cohomology.

If we only assume that the Frölicher spectral sequence of $X$ degenerates at $E_1$, is there a natural way to treat $H^{p,q}_{\bar\partial}(X)$ as a subspace of $H^k(X,\mathbb C)$? or is there a natural map $f:H^{p,q}_{\bar\partial}(X)\to H^k(X,\mathbb C)$ which is injective?

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