  In this note, we introduce a new type of positivity condition for the
curvature of a Hermitian manifold, which generalizes the notion of nonnegative
quadratic orthogonal bisectional curvature to the non-K\"ahler case. We derive
a Bochner formula for closed $(1, 1)$-forms from which this condition appears
naturally and prove that if a Hermitian manifold satisfy our positivity
condition, then any class $\alpha \in H^{1, 1}_{BC}(X)$ can be represented by a
closed $(1, 1)$-form which is parallel with respect to the Bismut connection.
Lastly, we show that such a curvature positivity condition holds on certain
generalized Hopf manifolds and on certain Vaisman manifolds.
