Let $R$ be a discrete valuation ring, $m$ the maximal ideal and $f:X \to \mathrm{Spec}(R)$ be a flat, proper morphism of relative dimension $1$. Assume further that $X$ is regular. For any $n>0$, denote by $X_n:=X \times_{\mathrm{Spec}(R)} \mathrm{Spec}(R/m^n)$ and by $i_n:X_n \hookrightarrow X$ the natural closed immersion.
Let $\mathcal{L}$ be an invertible sheaf on $X$. Suppose $\mathcal{L}'$ is another invertible sheaf on $X$ satisfying the property: for all $n>0$, $i_n^*\mathcal{L} \cong i_n^*\mathcal{L}'$. Is $\mathcal{L} \cong \mathcal{L}'$? Any reference which deals with similar questions will be also very welcome.