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?