I wrote up some notes on this in 2004.  There have been some developments since then that I will indicate below.

Denote the smooth, proper morphism as follows, $$\pi:\mathcal{X}\to \text{Spec}\ R.$$  Denote by $\omega_\pi$ the relative dualizing sheaf.  Since $\pi$ is flat and proper, also the fiber product morphism, $$\mathbb{P}^n\times_{\text{Spec}\ R}\mathcal{X}\to \text{Spec}\ R,$$ is also flat and proper.  For a flat, proper morphism, Michael Artin proved representability of the relative Hilbert functor by an algebraic space that is locally finitely presented and separated over the base.  Thus, there exists a universal pair $$(\rho:I\to \text{Spec}\ R,\phi:I\times_{\text{Spec}\ R}\mathbb{P}^n \xrightarrow{\cong} I\times_{\text{Spec}\ R}\mathcal{X})$$ of a separated, locally finitely presented scheme $I$ over $\text{Spec}\ R$ and an isomorphism $\phi$ of $I$-schemes, and this is compatible with arbitrary base change of $\text{Spec}\ R$. 

<B>Question 1.</B>  Is $\rho$ surjective on points?

This can be checked after surjective base change of $\text{Spec}\ R$.  Thus, without loss of generality, assume that the generic fiber $X_K$ is $K$-isomorphic to $\mathbb{P}^n_K$.  Let $H\subset X_K$ be an effective Cartier divisor whose associated invertible sheaf generates the Picard group.  Since $\mathcal{X}$ is smooth over $\text{Spec}\ R$, also $\mathcal{X}$ is regular.  Thus the closure $\mathcal{H}$ of $H$ in $\mathcal{X}$ is a Cartier divisor that is flat over $\text{Spec}\ R$.  Denote by $\mathcal{L}$ the associated invertible sheaf (the dual of the ideal sheaf of $\mathcal{H}$).  The restriction of $\mathcal{L}$ to $X_K$ is an ample generator of $\text{Pic}(X_K)$.

<B>Lemma 2.</B> Let $\pi:\mathcal{X}\to \text{Spec}\ R$ be a proper, flat morphism of schemes, and let $\mathcal{L}$ be an invertible sheaf on $\mathcal{X}$ whose restriction to $X_K$ is a generator of the Picard group.  If the closed fiber $X_k$ is integral, then every invertible sheaf on $\mathcal{X}$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$.

<B>Proof.</B>  Since $\pi$ is proper and flat with integral closed fiber, also $\mathcal{X}$ is integral.  Thus, every invertible sheaf on $\mathcal{X}$ is $R$-flat.  So the pushforward by $\pi$ is a finitely generated, torsion-free $R$-module, i.e., it is a finite, free $R$-module.

For every invertible sheaf $\mathcal{M}$ on $\mathcal{X}$, since $\mathcal{L}|_{X_K}$ generates $\text{Pic}(X_K)$, there exists an integer $m$ such that $\mathcal{M}|_{X_K}$ is isomorphic to $\mathcal{L}^{\otimes m}|_{X_K}$.  The pushforward $\pi_*\textit{Hom}_{\mathcal{O}_X}(\mathcal{M},\mathcal{L}^{\otimes m})$ is a finite, free $R$-module whose $K$-fiber equals $H^0(X_K,\mathcal{O}_{X_K})$.  Thus, there exists an element of this $R$-module whose $K$-germ is an invertible element of $H^0(X_K,\mathcal{O}_{X_K})$.  This element corresponds to a homomorphism of invertible sheaves, $$\alpha:\mathcal{M} \to \mathcal{L}^{\otimes m},$$ that is an isomorphism on $X_K$.  Thus, the support of the cokernel of $\alpha$ is an effective Cartier divisor on $\mathcal{X}$ that is disjoint from $X_K$.  

Since the closed fiber $X_k$ is integral, the support of the cokernel equals $r\underline{X}_k$ for some integer $r\geq 0$.  Thus $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}(-r\underline{X}_k)$ for some integer $r$.  Since $\mathcal{O}(-\underline{X}_k)$ is the pullback by $\pi$ of the maximal ideal of $R$, and since this is a free $R$-module, also $\mathcal{O}(-\underline{X}_k)$ is a free $\mathcal{O}_{\mathcal{X}}$-module.  Therefore, $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}$. <B>QED</B>

<B>Proposition 3.</B> With the same hypotheses as in Lemma 1, assume also that $\mathcal{X}$ is regular.  Then either the invertible sheaf $\mathcal{L}$ is $\pi$-ample or the invertible sheaf $\mathcal{L}^\vee$ is $\pi$-ample.

<B>Proof.</B>  This is automatic if $\pi$ is finite.  Hence assume that $\pi$ has positive fiber dimension.  By Chow's Lemma, a blowing up of $X_K$ is projective.  This blowing up is covered by complete intersection curves whose pushforwards to $X_K$ are numerically equivalent.  Up to replacing $\mathcal{L}$ by $\mathcal{L}^\vee$, assume that the degree of $\mathcal{L}$ on these curves is nonnegative.  The claim is that for every integer $m$ that is sufficient positive and divisible, the invertible sheaf $\mathcal{L}^{\otimes m}$ is globally generated and the complete linear system defines a finite morphism to projective space.

For every open affine $U \subset \mathcal{X}$ whose closed fiber $U_k$ is nonempty, for every pair $p,q\in U_k$ of distinct closed points, there exists $f\in H^0(U,\mathcal{O}_{\mathcal{X}})$ that is zero on $p$ and that is nonzero on $q$.  The zero scheme $\text{Zero}(f)$ is a Cartier divisor in $U$.  The closure in $\mathcal{X}$ of $\text{Zero}(f)$ is a Cartier divisor $D$ that contains $p$, yet does not contain $q$.  By the previous lemma, $\mathcal{O}(\underline{D})$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$.  Since $D$ is effective, its restriction to $X_K$ is effective.  Thus, the intersection number with the covering family of curves is positive.  Therefore, the integer $m$ is also positive.  Via this isomorphism, there exists a global section $s$ of $\mathcal{L}^{\otimes m}$ that vanishes at $p$ yet does not vanish at $q$.  Note that for every positive integer $d$, also $s^d$ is a global section of $\mathcal{L}^{\otimes md}$ that vanishes at $p$ yet does not vanish at $q$.  

Thus, for every integer $d\geq 1$, for the base locus $B_d$ of the complete linear system of $\mathcal{L}^{\otimes d}$, for every $q\in B_d$, there exists an integer $m\geq 1$ such that $B_{dm}$ is contained in $B_d$ yet does not contain $q$.  By Noetherian induction, for all integers $m$ that are sufficiently positive and divisible, $B_m$ is empty.  Moreover, for every closed point $p\in X_k$, for every open affine $U$ neighborhood of $p$, for every closed point $q\in U_k$, there exists an integer $m\geq 1$ such that the associated $R$-morphism $$\phi_m:\mathcal{X}\to \mathbb{P}H^0(\mathcal{X},\mathcal{L}^{\otimes m}),$$ does not contain $(p,q)$ in the fiber product $$\mathcal{X}\times_{\mathbb{P}H^0(\mathcal{L}^m)}\mathcal{X}\subset \mathcal{X}\times_{\text{Spec}\ R}\mathcal{X}.$$  Thus, by another Noetherian induction argument, for all integers $m$ that are sufficiently positive and divisible, every irreducible component of the fiber product projects finitely to $\mathcal{X}$ under projection on the first factor.  Thus, $\phi_m$ is a finite morphism, and $\mathcal{L}^{\otimes m}$ is ample. <B>QED</B>

<B>Nota bene.</B> There is a vast generalization of Lemma 2 for specializations of rationally connected varieties in the following article of Gounelas and Javanpeykar.

Frank Gounelas, Ariyan Javanpeykar<br>
Invariants of Fano varieties in families <br>
https://arxiv.org/abs/1703.0573

<B>Corollary 4.</B> With the same hypotheses as in Proposition 3, for an $R$-flat closed subscheme $\mathcal{Y}\subset \mathcal{X}$ of pure relative dimension $e$, if the $\mathcal{L}$-Hilbert polynomial of $\mathcal{Y}$ equals $P_{\mathcal{Y},\mathcal{L}}(t) = 1\cdot (t^e/e!) + O(t^{e-1})$, then the closed fiber $Y_k$ of the normalization of $\mathcal{Y}$ is geometrically integral.

<B>Proof.</B>  Since $\mathcal{Y}$ is $R$-flat, the Hilbert polynomial of the closed fiber equals the Hilbert polynomial of the generic fiber.  Since the normalization of $\mathcal{Y}$ is $S_2$, the closed fiber $Y_k$ is $S_1$.  Thus, to prove that $Y_k$ is geometrically integral, it suffices to prove that it is geometrically irreducible and reduced at the generic point.  The leading coefficient of the Hilbert polynomial is additive for irreducible components and multiplicative for multiplicity at a generic point.  Since the coefficient equals $1$, and since $\mathcal{L}$ is ample (so that the leading coefficient of every irreducible component with its reduced structure is a positive integer), it follows that $Y_k$ is geometrically irreducible and has multiplicity $1$ at every generic point. <B>QED</B>

<B>I will complete the argument soon.</B>  The basic idea now is to use specializations of lines to prove that $\mathcal{H}$ is smooth over $\text{Spec}\ R$ (or else the multiplicity at a singular point would give an intersection number with a specialization of a line that is greater than one).  Then, by an induction hypothesis, the scheme $\mathcal{H}$ is a projective space bundle over $\text{Spec}\ R$.  Finally, fixing a $k$-point $q$ of $X_k$ that is not contained in $H_k$, the specializations in $X_k$ of lines that contain $q$ each intersect $H_k$ in a unique point.  From this, it follows that the blowing up of $X_k$ along $q$ is isomorphic to the projective bundle over $H_k$ of $\mathcal{O}_{H_k}\oplus \mathcal{L}|_{H_k}$.  Using the induction hypothesis, this equals the blowing up of $\mathbb{P}^n_k$ along a $k$-point.