The proof follows mainly the arguments in [GK15, Proposition 6.4]. By Lemma 3.3 and Lemma 3.4, we may assume that $K$ is algebraically closed and that $V$ is reduced. Let $({\mathfrak{V}},{\mathfrak{L}})$ be a formal ${K^{\circ}}$-model of $(V,L)$ with ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{\mathfrak{L}}$. We assume that ${\|\hskip 4.30554pt\|}$ is semipositive in every $x\in V$. We choose a closed curve $C$ in ${\mathfrak{V}}_{s}$ which is proper over ${\tilde{K}}$. We have to show that $\deg_{\mathfrak{L}}(C)\geq 0$. By surjectivity of the reduction map $\pi:V\to{\mathfrak{V}}_{s}$, there is $x\in V$ such that $\pi(x)$ is the generic point of $C$. Since ${\|\hskip 4.30554pt\|}$ is semipositive in $x$, there is a compact strictly $K$-affinoid neighborhood $W$ of $x$ and a nef formal ${K^{\circ}}$-model $({\mathfrak{W}},{\mathfrak{M}})$ of $(W,L|_{W})$ such that ${\|\hskip 4.30554pt\|}={\|\hskip 4.30554pt\|}_{\mathfrak{M}}$ over $W$. Using Proposition 3.5, we may always replace the models ${\mathfrak{W}}$ and ${\mathfrak{V}}$ by dominating formal ${K^{\circ}}$-models and the line bundles ${\mathfrak{M}}$ and ${\mathfrak{L}}$ by their pull-backs. By [BL93b, Corollary 5.4], we may therefore assume that ${\mathfrak{W}}$ is a formal open subset of ${\mathfrak{V}}$. Then ${\mathfrak{L}}|_{\mathfrak{W}}$ is also a formal ${K^{\circ}}$-model of $L|_{W}$ and hence Proposition 3.5 shows that ${\mathfrak{L}}|_{\mathfrak{W}}$ is nef.

Since $W$ is a neighbourhood of $x$ and since $V$ is boundaryless, we conclude that the boundary of $W$ is the topological boundary of $W$ in $V$ (see [Ber90, Corollary 2.5.13(ii), Proposition 3.1.3(ii)]). In particular, $x$ is no boundary point of $W$ as $W$ is a neighborhood of $x$. Using [CD12, Lemma 6.5.1], such interior points are characterized by the property that the closure of the reduction in ${\mathfrak{W}}_{s}$ is proper over ${\tilde{K}}$. We conclude that the closure of $\pi(x)$ in ${\mathfrak{W}}_{s}$ is equal to $C$. Since ${\mathfrak{L}}|_{\mathfrak{W}}$ is nef, it follows that $\deg_{\mathfrak{L}}(C)\geq 0$. ∎