A note on Griffiths' conjecture in rank $3$

Let $ (E,h) $ be a Griffiths semipositive Hermitian holomorphic vector bundle of rank $ 3 $ over a complex manifold. The purpose of this paper is to show the positivity of the characteristic differential forms $ c_2(E,h) $ and $ c_1(E,h) \wedge c_2(E,h) - c_3(E,h) $, which arise from the Chern curvature of the given vector bundle. Such result provides new evidences towards the well-known Griffiths' conjecture about the positivity of the Schur polynomials in the Chern forms of Griffiths (semi)positive vector bundles. The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.