  The Griffiths conjecture asserts that every ample vector bundle $E$ over a
compact complex manifold $S$ admits a hermitian metric with positive curvature
in the sense of Griffiths. In this article we give a sufficient condition for a
positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce a
Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests
to study the relative K\"ahler-Ricci flow on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$
for the fibration $\mathbb{P}(E^*)\to S$. We define a flow and give arguments
for the convergence.
