Griffiths positivity versus ampleness

Prove that every ample holomorphic vector bundle can be endowed with a Griffiths-positive Hermitian metric, thereby establishing the equivalence between ampleness and Griffiths positivity for vector bundles.

Background

Griffiths positivity is a differential-geometric positivity notion for Hermitian holomorphic vector bundles, while ampleness is its algebro-geometric counterpart. The two notions are equivalent for line bundles and for vector bundles over curves, but the equivalence for general ample vector bundles remains unresolved. The paper notes that constructing a Griffiths-positive metric on a general ample vector bundle is difficult, although a strategy due to Demailly has been proposed.