The bounded-geometry characterization by the five gradients inequality

Let $({\mathcal{M}},g)$ be a complete Riemannian manifold. It is of bounded geometry when its curvature tensor and all its covariant derivatives are bounded.

Bounded-geometry conjecture. The manifold $({\mathcal{M}},g)$ is of bounded geometry if and only if the five gradients inequality holds.

The equivalence is known for compact Riemannian manifolds, while the general characterization of bounded geometry remains open.