Let $R$ be a discrete valuation ring which is defined geometrically by a $d$-dimensional normal variety $B$ over the field $k$ (as in Remark 7.2). Let $\mathscr{X}_{R}$ be a projective integral scheme over $R$ with $n$-dimensional regular generic fiber $X^{\prime}\coloneqq\mathscr{X}_{R,\eta}$. We assume that resolution of singularities holds over $k$ in dimension $d+n$. Then for any ample line bundle $L^{\prime}$ on $X^{\prime}$, there exists $m\in\mathbb{N}_{>0}$ and an ample extension $\mathscr{L}_{R}^{\prime}$ of $(L^{\prime})^{\otimes m}$ to a regular $R$-model $\mathscr{X}_{R}^{\prime}$ of $X^{\prime}$ with a projective morphism $\mathscr{X}_{R}^{\prime}\to\mathscr{X}_{R}$ over $R$ extending the identity on $X^{\prime}$.