Since this is stated here under more general assumptions than in [GK15, Proposition 5.11], we sketch the argument. Let $({\mathfrak{W}},{\mathfrak{L}})$ be the ${K^{\circ}}$-model for the given formal metric on $L|_{W}$. We may assume that ${\mathfrak{W}}$ is a formal open subset of a formal ${K^{\circ}}$-model ${\mathfrak{V}}$ of $V$ [Bos14, Lemma 8.4.5]. By the argument in [BL93a, Lemma 5.7], there is a coherent ${\mathcal{O}}_{\mathfrak{V}}$-module ${\mathscr{F}}$ on ${\mathfrak{V}}$ which extends ${\mathfrak{L}}$. This works even for paracompact $V$ as noted in the proof of [CD12, Proposition 6.2.13] and the argument there (or in the proof of [Gub98, Lemma 7.6]) shows that after replacing ${\mathfrak{V}}$ by a suitable admissible blowing-up, we may assume that ${\mathscr{F}}$ is a line bundle. Then the associated formal metric satisfies the claim. ∎