Counterexamples of positivity implications in Kähler curvature tensors

It is well-known that the curvature properties of Kähler metrics 'flow downward'. To give meaning to this statement, let us say that a tensor \( \alpha \) dominates a tensor \( \beta \) if any of the following implications hold: \begin{align*} \alpha > 0 &\implies \beta > 0, & \alpha &\geq 0 \implies \beta \geq 0, \\ \alpha < 0 &\implies \beta < 0, & \alpha &\leq 0 \implies \beta \leq 0.…