Griffiths positivity does not imply positivity of the top Chern form

Abstract: For every integer $r\geq9$, 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.