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 Abstract:   Ricci flow is know to preserve many natural curvature positivity conditions. These include positivity of the Ricci curvature in dimension 3, positivity of the curvature operator, positivity of the bisectional holomorphic curvature in the Kahler setting. In 2009 Streets and Tian have introduced a family of metric flows on general Hermitian manifolds, generalizing the Kahler-Ricci flow. We prove that for a specific flow in this family many natural curvature conditions are preserved. This result gives a possible approach to a weak differential-geometrical version of Campana-Peternell conjecture, which in its original form states that a Fano manifold with nef tangent bundle is rational homogeneous.