Obstructions to deformations of d.g. modules

Let $\mathbf{k}$ be a field and $n\geq1$. There exist a differential graded $\mathbf{k}$-module $(V,d)$ and various approximations to a differential $d+td_{1}+t^{2}d_{2}+\cdots+ t^{n}d_{n}$ on $V[[t]],$ one of which gives a non-trivial deformation, another is obstructed, and another is unobstructed at order $n$. The analogous problem in the category of $\mathbf{k}$-algebras in characteristic zero remains a long-standing open question.