It follows from our assumptions that $X$ is base change of the generic fiber $\mathscr{X}_{R,\eta}$ of $\mathscr{X}_{R}/R$ to $K$. Since $X$ is smooth, we conclude that $\mathscr{X}_{R,\eta}$ is smooth as well [EGAIV, Cor. 17.7.3]. By Proposition 2.9(vii), it is enough to prove the claim for any positive multiple of $\theta$. Using this and Lemma 7.5 below, we see that by passing to dominant models, we may assume that $\mathscr{X}_{R}$ is regular and that the restriction of $\mathscr{L}_{R}$ to $\mathscr{X}_{R,\eta}$ extends to an ample line bundle on $\mathscr{X}_{R}$. By Remark 7.2, these conditions are equivalent to Assumption 7.1 and hence the claim follows from Theorem 7.3. ∎