Liftings of point-wise finite dimensional persistence modules over local commutative Artinian rings

Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$ with residue field isomorphic to $\mathbf{k}$, there is a generalized persistence module $M: \mathscr{C} \to R\textup{-Mod}$, such that for all $x\in \mathrm{Ob}(\mathscr{C})$, $M(x)$ is free over $R$ with finite rank and $\mathbf{k}\otimes_R M(x)\cong V(x)$. If $V$ is a direct sum of indecomposable persistence modules $V_I: \mathscr{C}\to \mathbf{k}\textup{-Mod}$ with endomorphism ring isomorphic to $\mathbf{k}$, then $M$ is a direct sum of indecomposables $M_I:\mathscr{C}\to R\textup{-Mod}$ with endomorphism ring isomorphic to $R$