We remark that ${\mathfrak{V}}_{s}\otimes_{\tilde{K}}\tilde{L}\simeq({\mathfrak{V}}\hat{\otimes}_{{K^{\circ}}}L^{\circ})_{s}$. Hence the result follows from the fact that a nef line bundle $\mathcal{L}$ on a proper variety over $\tilde{K}$ remains nef after pull back to $\tilde{F}$. This is proven in the projective case in [EFM, Remark 1.3.25] and the proper case follows from Chow’s lemma and the projection formula. ∎