The proof proceeds in three steps. First, we use a result of Lütkebohmert about vertical blowing ups to show that $L^{\prime}$ may be assumed to extend to an ample line bundle $\mathscr{H}$ on $\mathscr{X}_{R}$. In a second step, we show that $\mathscr{X}_{R}$ may be also assumed to be semi-factorial by a theorem of Pépin. In a third step, we use resolution of singularities to construct our desired regular model $\mathscr{X}_{R}^{\prime}$.

Step 1: Replacing $L^{\prime}$ by a positive tensor power, we may assume that $L^{\prime}$ has an ample extension $\mathscr{H}_{R}$ to a projective $R$-model ${\mathscr{Y}}_{R}$. There is a blow up $\pi:\mathscr{Z}_{R}\to{\mathscr{Y}}_{R}$ in an ideal sheaf $\mathcal{J}$ supported in the special fiber of ${\mathscr{Y}}_{R}$ such that the identity on $X^{\prime}$ extends to a morphism $\mathscr{Z}_{R}\to\mathscr{X}_{R}$ [Lü93, Lemma 2.2]. Then $\pi^{-1}(\mathcal{J})=\mathcal{O}_{\mathscr{Z}_{R}/{\mathscr{Y}}_{R}}(1)$ and hence there is $\ell\in\mathbb{N}_{>0}$ such that $\pi^{*}(\mathscr{H}^{\otimes\ell})\otimes\mathcal{O}_{\mathscr{Z}_{R}/{\mathscr{Y}}_{R}}(1)$ is ample [Har77, Prop. II.7.10]. We conclude that by replacing $\mathscr{X}_{R}$ by $\mathscr{Z}_{R}$ and by passing to a positive tensor power of $L^{\prime}$, we may assume that $L^{\prime}$ has an ample extension $\mathscr{H}_{R}$ to $\mathscr{X}_{R}$. This completes the first step.

Step 2: By a result of Pépin [Pé13, Thm. 3.1], there is a a blowing-up morphism $\pi^{\prime}\colon\mathscr{Z}_{R}^{\prime}\to\mathscr{X}_{R}$ centered in the special fiber of $\mathscr{X}_{R}$ such that $\mathscr{Z}_{R}^{\prime}$ is semi-factorial. The latter means that every line bundle on the generic fiber $\mathscr{Z}_{R,\eta}^{\prime}$ of $\mathscr{Z}_{R}^{\prime}$ over $R$ extends to a line bundle on $\mathscr{Z}_{R}^{\prime}$. Similarly as in the first step, we may assume that a positive tensor power of $L^{\prime}$ extends to an ample line bundle on $\mathscr{Z}_{R}^{\prime}$. Replacing $\mathscr{X}_{R}$ by $\mathscr{Z}_{R}^{\prime}$ and $L^{\prime}$ by this positive tensor power, we get the second step.

Step 3: We may assume that $B$ is affine. Using $R=\mathcal{O}_{B,b}$ for some $b\in B^{(1)}$, it is clear that $\mathscr{X}_{R}$ extends to a projective integral scheme $\mathscr{X}_{B}$ over $B$. By using resolution of singularities over $k$ in dimension $d+n$, there is a regular integral scheme $\mathscr{X}_{B}^{\prime}$ and a projective morphism $\varphi_{B}\colon\mathscr{X}_{B}^{\prime}\to\mathscr{X}_{B}$ which is an isomorphism over the regular locus of $\mathscr{X}_{B}$. Since $X^{\prime}$ is contained in the regular locus of $\mathscr{X}_{B}$, we conclude that $\varphi_{B}$ maps the generic fiber $\mathscr{X}_{B,\eta}^{\prime}$ of $\mathscr{X}_{B}^{\prime}$ over $B$ isomorphically onto $X^{\prime}=\mathscr{X}_{B,\eta}$. As usual, we read this isomorphism as an identification. Then we get an induced projective morphism

extending the identity on $X^{\prime}$. The same argument as in the first step gives $m\in\mathbb{N}_{>0}$ such that

is an ample line bundle on $\mathscr{X}_{R}^{\prime}$. Let $F$ be the restriction of $\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)$ to $\mathscr{X}_{R,\eta}^{\prime}=\mathscr{X}_{R,\eta}=X^{\prime}$. Then $\mathscr{L}_{R}^{\prime}$ is a model of $(L^{\prime})^{\otimes m}\otimes F$. To prove the lemma, we have to ensure that $F$ may be assumed to be $\mathcal{O}_{X^{\prime}}$. To do so, we use that $\mathscr{X}_{R}$ is semi-factorial to extend $F$ to a line bundle $\mathscr{F}_{B}$ on $\mathscr{X}_{R}$. Then we may replace $\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)$ by $\mathcal{O}_{\mathscr{X}_{R}^{\prime}/\mathscr{X}_{R}}(1)\otimes\varphi_{R}^{*}(\mathscr{F}^{-1})$ to deduce the claim. ∎