A note on pseudo-effective vector bundles with vanishing first Chern number over non-Kähler manifolds

In this note, We show that over a compact Hermitian manifold $(X, \omega )$ whose metric satisfies $\partial \bar{\partial }\omega ^{n - 1} = \partial \bar{\partial }\omega ^{n - 2} = 0$, every pseudo-effective vector bundle with vanishing first Chern number is in fact a numerically flat vector bundle