A characterization of Griffiths semipositivity by top-degree optimal $\bar{\mathrm{\partial }}$-estimates

We prove that a singular Hermitian metric $h$ on a vector bundle agrees almost everywhere with a Griffiths semipositive metric if its dual metric $h^{\ast }$ is locally integrable and the $\bar{\mathrm{\partial }}$-equation admits solutions with optimal $L^2$-estimates for $(n,n)$-forms, where $n$ is the dimension of the base manifold. In particular, under this integrability assumption, Nakano semipositivity in the sense of optimal estimates implies Griffiths semipositivity.