Signed Euler characteristic under nonpositive bisectional curvature

Prove that an n-dimensional compact Kähler manifold with nonpositive bisectional curvature and quasi-negative Ricci curvature has positive signed Euler characteristic, namely (-1)^n c_n[M] > 0.

Background

This conjecture is presented as a complex analogue of the Hopf conjecture. The paper establishes that, for compact Kähler manifolds with nonpositive bisectional curvature, the conjecture is equivalent to simultaneous positivity of the relevant Chern numbers. It is known in complex dimension two and in dimensions at most four under the stronger assumption of nonpositive Riemannian sectional curvature, but remains unresolved in its full stated generality.