We make use of the universal connection on the bundle of principal connections; the bundle structure is governed by the action of the group on the first jet bundle. Each section determines a connection in the principal bundle, which in case of the frame bundle allows a metric completion projecting onto the corresponding b-completion. It is shown that b-incompleteness of the base manifold is stable under perturbations of the chosen section. Hence, for instance, for (pseudo) Riemannian manifolds including spacetimes, b-incompliteness is a stable condition under conformal deformations of the metric. This reinforces the belief that general relativistic singularities cannot be removed by quantization.