Let $R$ be a local Noetherian domain with fraction field $K$ and residue field $\Bbbk$. Let $C^{\bullet}$ be a bounded complex of free, finitely generated $R$-modules. Suppose that $C^{\bullet} \otimes_R K$ and $C^{\bullet} \otimes_R \Bbbk$ are both exact. Does it follow that $C^{\bullet}$ is exact?

[Note: The converse holds, since tensoring anything by an exact sequence of flat modules (with zeros on both ends) gives an exact sequence.]