[proofplan]
The proof is the positivity consequence of the assumed Berndtsson curvature formula. Fix a Nakano test vector $(\xi_j,u_j)$. The formula writes the contracted curvature pairing as a sum of two terms. The fibre-integral term is nonnegative because $i\Theta_h(L)$ is a semipositive Hermitian form and the forms $u_j$ contribute a pointwise rank-one positive semidefinite density. The Green-operator term is nonnegative because $N_{y_0}$ is positive self-adjoint. Since every Nakano test vector gives a nonnegative contracted curvature pairing, the Chern curvature of the direct image bundle is Nakano semipositive.
[/proofplan]
[step:Fix an arbitrary Nakano test vector]
Fix $y_0\in Y$, tangent vectors
\begin{align*}
\xi_1,\dots,\xi_r\in T_{y_0}Y,
\end{align*}
and fibrewise sections
\begin{align*}
u_1,\dots,u_r\in E_{y_0}=H^0(X_{y_0},K_{X_{y_0}}\otimes L|_{X_{y_0}}).
\end{align*}
Nakano semipositivity of $E$ is exactly the assertion that
\begin{align*}
\sum_{j,k=1}^r
\big\langle \Theta^E(\xi_j,\overline{\xi_k})u_j,u_k\big\rangle_{y_0}\ge 0
\end{align*}
for every such tuple. It is therefore enough to prove this inequality for the fixed tuple.
[/step]
[step:Apply the assumed direct-image curvature formula]
By the assumed Berndtsson curvature formula, the contracted curvature pairing for this tuple is
\begin{align*}
\sum_{j,k=1}^r
\big\langle \Theta^E(\xi_j,\overline{\xi_k})u_j,u_k\big\rangle_{y_0}
&=
c_n\int_{X_{y_0}}
\sum_{j,k=1}^r
\big(i\Theta_h(L)\big)(V_j,\overline{V_k})\,
u_j\wedge\overline{u_k}\,e^{-\varphi}
\\
&\qquad
+\big\langle N_{y_0}\mu,\mu\big\rangle_{L^2(X_{y_0})}.
\end{align*}
Thus it remains only to check that the two terms on the right are nonnegative.
[/step]
[step:Show the curvature integral is nonnegative]
Let $x\in X_{y_0}$. Since $i\Theta_h(L)$ is semipositive as a Hermitian form on $T_xX$, the matrix
\begin{align*}
A(x)=\Big(\big(i\Theta_h(L)\big)(V_j(x),\overline{V_k(x)})\Big)_{j,k=1}^r
\end{align*}
is positive semidefinite.
Choose local coordinates $z=(z_1,\dots,z_n)$ on the fibre and a local holomorphic frame $e$ for $L$ near $x$. Then each fibrewise $L$-valued $(n,0)$-form can be written locally as
\begin{align*}
u_j=f_j\,dz_1\wedge\cdots\wedge dz_n\otimes e.
\end{align*}
With respect to the positive density determined by the fibre coordinates and the metric $h$, the integrand in the first term is locally
\begin{align*}
\sum_{j,k=1}^r A_{j\bar k}(x) f_j(x)\overline{f_k(x)}
\end{align*}
times that positive density. Because $A(x)$ is positive semidefinite, this scalar is nonnegative at every point $x$. Therefore
\begin{align*}
c_n\int_{X_{y_0}}
\sum_{j,k=1}^r
\big(i\Theta_h(L)\big)(V_j,\overline{V_k})\,
u_j\wedge\overline{u_k}\,e^{-\varphi}\ge 0.
\end{align*}
[/step]
[step:Show the Green-operator term is nonnegative]
By hypothesis, $N_{y_0}$ is a positive self-adjoint Green operator for the fibrewise $\bar\partial$-Laplacian. Hence for every form $\alpha$ in its domain,
\begin{align*}
\langle N_{y_0}\alpha,\alpha\rangle_{L^2(X_{y_0})}\ge 0.
\end{align*}
Applying this to $\alpha=\mu$ gives
\begin{align*}
\big\langle N_{y_0}\mu,\mu\big\rangle_{L^2(X_{y_0})}\ge 0.
\end{align*}
[/step]
[step:Conclude Nakano semipositivity]
The curvature formula expresses the Nakano curvature pairing as the sum of the nonnegative curvature-integral term and the nonnegative Green-operator term. Therefore
\begin{align*}
\sum_{j,k=1}^r
\big\langle \Theta^E(\xi_j,\overline{\xi_k})u_j,u_k\big\rangle_{y_0}\ge 0.
\end{align*}
The point $y_0$, tangent vectors $\xi_j$, and fibrewise sections $u_j$ were arbitrary. Hence the Chern curvature of $(E,\langle\cdot,\cdot\rangle)$ is Nakano semipositive.
[/step]
