Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

We give counterexamples to Griffiths' conjecture in rank two on the quadric surface ${\mathbb{P}}^1\times {\mathbb{P}}^1$. We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with $\mathcal{O}(1,1)$, are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.