Singular Nakano positivity of direct image sheaves of adjoint bundles

In this paper, we consider a proper Kähler fibration $f\text{\hspace{0.17em}⁣}:X\to Y$ and a singular Hermitian line bundle $(L,h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{\ast }({\mathcal{O}}_X(K_{X\mathrm{/}Y}+L)\otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\bar{\mathrm{\partial }}$-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.