Griffiths positivity does not imply positivity of the top Chern form

For every integer $r\ge 9$, we construct a smooth Griffiths-positive Hermitian metric on ${\mathcal{O}}_{{\mathbb{P}}^r}(1{)}^{\oplus r}$ whose top Chern form is pointwise negative on a nonempty open set. This gives counterexamples on compact projective manifolds to Griffiths’ conjecture on the positivity of Chern–Weil forms.