Completeness from complete vector fields for coframings

Let $M$ be an $n$-dimensional manifold with a coframing, and let its canonical metric be the Riemannian metric induced by the coframing. Consider the Lie algebra of vector fields generated by the coframing. Completeness conjecture. If this Lie algebra consists entirely of complete vector fields, then the canonical metric of the coframing is complete. The claim proposes a sufficient condition for metric completeness of a coframed manifold; the supplied text gives no resolution.