Let $S $ be the spectrum of a complete dvr with algebraically closed residue field. Let $\eta$ be its generic point and let $s$ be its closed point with $k(s)$ of positive characteristic. Let $X\to S$ be a smooth proper morphism of schemes whose geometric fibres are connected.

Edit:For all $b$ in $S$, assume that $\mathrm{H}^1(X_b,\mathcal{O}_{X_b}) = \mathrm{H}^0(X_b,\Omega^1_{X_b})=0$ and that $\mathrm{Pic}(X_b)$ is torsion-free. Moreover, assume that $\mathrm{rk}(\mathrm{Pic}(X_{\eta})) = \mathrm{rk}(\mathrm{Pic}(X_s))$. Also, assume the fibres of $X\to S$ are Fano varieties. (Some of these assumptions are redundant.)

Let $L$ be a line bundle on $X_s$ such that $L^{\otimes p}\cong \omega_{X_s/k(s)}$, where $p$ equals the characteristic of $k(s)$.

Question.Does $L$ lift to $X$?

The motivation for this question comes from a question about the index of a Fano variety. Namely, it seems reasonable to suspect that the index of a Fano variety doesn't jump upon specialization. This is clear in characteristic zero, as $\mathrm{H}^1(X,\mathcal{O}_X)=\mathrm{H}^2(X,\mathcal{O}_X)=0$ for a Fano variety $X$ over $\mathbb C$. However, the vanishing of these cohomology groups is currently not known for Fano varieties over fields of positive characteristic. (The vanishing of $\mathrm{H}^1(X,\mathcal{O}_X)$ is known for SRC Fano varieties.)