Chern classes of ample bundles

In the article “Ample vector bundles”, Publ. Math., № 29, R. Hartshorne has extended the notion of ample vector bundle to vector bundles of arbitrary rank and has raised the following question. Let $\mathscr E$ be an ample vector bundle over a nonsingular algebraic variety $X$ and assume that the rank of $\mathscr E$ is equal to $n$. Is it true that the $i$th Chern class $c_i(\mathscr E)$ is numerically positive for $i\leqslant n$? In this paper it is proved that in the case $\operatorname{dim}X=2$ the degree of the point-cycle $c_2(\mathscr E)$ is positive.
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