On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n, and let E be an ample vector bundle over X. We show that any Schur class of E, lying in the cohomology group of bidegree (n − 1, n − 1), has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld (1983) for certain degrees.