By definition if ${\mathfrak{L}}$ is nef, then $\|\ \|_{\mathfrak{L}}$ is semipositive, so we only have to prove the reverse implication. Hence we assume that ${\|\hskip 4.30554pt\|}_{\mathfrak{L}}$ is a semipositive formal metric and we have to show that ${\mathfrak{L}}$ is nef. Using Lemma 3.3, we can replace $K$ by ${\mathbb{C}}_{K}$ and hence assume that $K$ is algebraically closed.

By definition of semipositivity, there is a nef ${K^{\circ}}$-model ${\mathfrak{M}}$ of $L$ on some model ${\mathfrak{W}}$ of $V$ with ${\|\hskip 4.30554pt\|}_{\mathfrak{L}}={\|\hskip 4.30554pt\|}_{\mathfrak{M}}$. There exists a model ${\mathfrak{X}}$ of $V$ which dominates both ${\mathfrak{V}}$ and ${\mathfrak{W}}$. Let $\pi:{\mathfrak{X}}\to{\mathfrak{V}}$ be the induced morphism. Since the induced morphism on the special fibers $\pi_{s}\colon{\mathfrak{X}}_{s}\to{\mathfrak{V}}_{s}$ is proper and surjective, by the projection formula, ${\mathfrak{L}}$ is nef if and only $\pi^{*}{\mathfrak{L}}$ is nef. Hence replacing $({\mathfrak{V}},{\mathfrak{L}})$ by $({\mathfrak{X}},\pi^{*}{\mathfrak{L}})$, we can assume that ${\mathfrak{V}}$ dominates ${\mathfrak{W}}$.

Let ${\mathfrak{V}}_{\rm red}$ be the reduced structure on ${\mathfrak{V}}$. Hence ${\mathfrak{V}}_{\rm red}\to{\mathfrak{V}}$ is finite. Locally, ${\mathfrak{V}}_{\rm red}$ is given by ${\rm Spf}(A)$ for some reduced admissible ${K^{\circ}}$-algebra. Let ${\mathscr{A}}\coloneqq A\otimes_{{K^{\circ}}}K$. It is a strictly $K$-affinoid algebra, and by [BGR84, 6.4.3] $A^{\prime}\coloneqq{\mathscr{A}}^{\circ}$ is an admissible $K^{\circ}$-algebra, and moreover $A\to A^{\prime}$ is finite and induces an isomorphism on the generic fibers. By [BGR84, 7.2.6 Proposition 3], we can glue the morphisms ${\rm Spf}(A^{\prime})\to{\rm Spf}(A)$ to get a model ${\mathfrak{V}}^{\prime}$ of $V_{\rm red}$ such that ${\mathfrak{V}}^{\prime}\to{\mathfrak{V}}_{\rm red}$ is finite. In particular, we deduce that the induced morphisms ${\mathfrak{V}}^{\prime}_{s}\to({\mathfrak{V}}_{\rm red})_{s}\to{\mathfrak{V}}_{s}$ are proper and surjective, and we conclude from the projection formula that ${\mathfrak{L}}$ is nef if and only if its pull back ${\mathfrak{L}}^{\prime}$ to ${\mathfrak{V}}^{\prime}$ is nef.

By construction, ${\mathfrak{V}}^{\prime}$ is locally of the form ${\rm Spf}({\mathscr{A}}^{\circ})$, hence we deduce that ${\mathfrak{V}}^{\prime}_{s}$ is locally given by ${\rm Spec}(\tilde{{\mathscr{A}}})$ which is reduced. Now we use the fact that on an admissible formal scheme with reduced special fibre and with $K$ algebraically closed, the metric ${\|\hskip 4.30554pt\|}_{{\mathfrak{L}}^{\prime}}$ determines the model ${\mathfrak{L}}^{\prime}$ up to isomorphism (see [Gub98, Proposition 7.5]). Using that ${\|\hskip 4.30554pt\|}_{{\mathfrak{M}}^{\prime}}={\|\hskip 4.30554pt\|}_{\mathfrak{L}}={\|\hskip 4.30554pt\|}_{{\mathfrak{L}}^{\prime}}$ for the pull-back ${\mathfrak{M}}^{\prime}$ of ${\mathfrak{M}}$ to ${\mathfrak{V}}^{\prime}$, we deduce that ${\mathfrak{M}}^{\prime}\cong{\mathfrak{L}}^{\prime}$. As above, the pull-back ${\mathfrak{M}}^{\prime}$ of ${\mathfrak{M}}$ is nef and hence ${\mathfrak{L}}^{\prime}$ is nef. ∎