Singular Nakano positivity of direct image sheaves of adjoint bundles

In this paper, we consider a proper K\"ahler fibration $f \colon X \to Y$ and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf.