Local smoothing conjecture for holomorphic sections and c9-trace positivity

Let $E$ be a holomorphic vector bundle with a singular Hermitian metric $h$, and let $c9$ be the Kähler metric from Proposition cref{smoothing of tr-c9 semi-posi for constant section}. A smooth Hermitian metric is c9-trace semi-positive when its c9-trace curvature is nonnegative. Local holomorphic-section approximation conjecture. Under the situation of that proposition, the section $c3$ should be allowed to be holomorphic rather than constant; equivalently, locally there should exist smooth Hermitian metrics $(h_bd)_{bdinmathbb{N}}$ that are c9-trace semi-positive and decrease to $h$. This asks whether the constant-section smoothing result extends to holomorphic sections, and the source presents it as a conjectural extension without giving a resolution.