# Does equality of Laplacians imply Kähler II?

For a compact Hermitian manifold, the metric being Kähler is equivalent the identity $$\Delta_{\partial} = \Delta_{\overline{\partial}},$$ where $$\Delta_{\partial}$$ and $$\Delta_{\overline{\partial}}$$ are the $$\partial$$ and $$\overline{\partial}$$-Laplacians. This is discussed in this question.

Is there an analogous result for the $$\mathrm{d}$$-Laplacian $$\Delta_{\mathrm{d}}$$? That is, does the identity $$\Delta_{\partial} = \Delta_{\mathrm{d}}$$ or the identity $$\Delta_{\overline{\partial}} = \Delta_{\mathrm{d}},$$ imply that an Hermitian metric is Kähler?