Coherence conjecture for multiplier ideal sheaves of Griffiths semi-positive metrics

Let $(E,h)$ be a holomorphic vector bundle over an $n$-dimensional complex manifold $X$ with a singular Hermitian metric $h$. Let $**\operatorname{Griffiths}$-semi-positive$**$ mean that $h$ has Griffiths semi-positive curvature. The higher rank analogue of the multiplier ideal sheaf, denoted by $\mathcal{E}(h)$, is coherent.

Coherence conjecture. If $h$ is Griffiths semi-positive, then $\mathcal{E}(h)$ is coherent.

Coherence is known in several cases, including metrics with stronger positivity such as Nakano positivity, but the Griffiths semi-positive case is posed as the natural question because Griffiths positivity is strictly weaker than Nakano positivity.