Simultaneous positivity for semipositive tangent or cotangent bundles

Determine whether every compact complex, Hermitian, or Kähler manifold whose holomorphic tangent bundle or cotangent bundle satisfies one of the semipositivity conditions of global generation, Bott–Chern semipositivity, Griffiths semipositivity, or nefness is simultaneously positive, meaning that all of its relevant Chern numbers or signed Chern numbers are either strictly positive or all vanish.

Background

The paper defines simultaneous positivity as the property that all relevant Chern numbers, or signed Chern numbers for cotangent-bundle settings, are simultaneously positive or simultaneously zero. Several classes are already known to have this property, including compact Kähler manifolds with globally generated cotangent bundle and manifolds with semipositive bisectional curvature. The question asks whether the phenomenon extends to the broader collection of semipositivity notions listed in the paper.