A converse of Berndtsson's theorem on the positivity of direct images

Berndtsson’s famous theorem asserts that, for a compact Kähler fibration $p:X\to Y$, the direct image bundle $p_{\ast }(K_{X\mathrm{/}Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson’s theorem in the case of a projective fibration: if $p_{\ast }(K_{X\mathrm{/}Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive.