Sign of $\int_XTr(F_h^2)$

Question

Let $(E,h)\to X$ be a holmorphic Hermitian vector bundle over a compact Kahler manifold. Denote by $F_h$ the curvature of its Chern connection. Can we know a priori the sign of the quantity $$\int_XTr(F_h^2)\wedge \omega^{n-2}$$
or if $X$ is a surface the sign of $$\int_XTr(F_h^2)$$