A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a non-Kahler compact complex manifold admits a hermitian metric such that the wedge product of two ${\bar{\partial}}$-harmonic forms is ${\bar{\partial}}$-harmonic?

A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a compact complex manifold admits a hermitian metric such that the wedge product of two ${\bar{\partial}}$-harmonic forms is ${\bar{\partial}}$-harmonic?