Every ${K^{\circ}}$-model of a projective variety $X$ is dominated by a projective ${K^{\circ}}$-model [Gub03, Proposition 10.5]. Hence we may assume that ${\mathscr{X}}_{0}$ is projective. There is $m_{1}\in{\mathbb{N}}$ and a closed immersion of $X$ into ${\mathbb{P}}_{K}^{N}$ such that $L^{\otimes m_{1}}={\mathcal{O}}_{{\mathbb{P}}_{K}^{N}}(1)|_{X}$. Then the closure of $X$ in ${\mathbb{P}}_{K^{\circ}}^{N}$ is a ${K^{\circ}}$-model ${\mathscr{X}}_{1}$ of $X$ which has an ample line bundle ${\mathscr{L}}_{1}$ such that ${\mathscr{L}}_{1}|_{X}=L^{\otimes m_{1}}$. Then the closure of the diagonal in ${\mathscr{X}}_{0}\times_{K^{\circ}}{\mathscr{X}}_{1}$ is a projective ${K^{\circ}}$-model ${\mathscr{X}}$ of $X$ and the canonical projection $p_{1}:{\mathscr{X}}\to{\mathscr{X}}_{1}$ is a projective morphism, hence there is a closed immersion of ${\mathscr{X}}$ into a projective space ${\mathbb{P}}_{{\mathscr{X}}_{1}}^{k}$ over ${\mathscr{X}}_{1}$. Let ${\mathscr{E}}$ be the restriction of ${\mathcal{O}}_{{\mathbb{P}}_{{\mathscr{X}}_{1}}^{k}}(1)$ to ${\mathscr{X}}$. Since ${\mathscr{E}}$ is relatively ample with respect to $p_{1}$ and since ${\mathscr{L}}_{1}$ is an ample line bundle on ${\mathscr{X}}_{1}$, there is $m_{2}\in{\mathbb{N}}$ such that ${\mathscr{L}}:=p_{1}^{*}({\mathscr{L}}_{1})^{\otimes m_{2}}\otimes{\mathscr{E}}$ is ample on ${\mathscr{X}}$. Then ${\mathscr{L}}$ is a ${K^{\circ}}$-model of $L^{\otimes m}$ for $m:=m_{1}m_{2}$. ∎