A condition for the stability of $\mathbb{R}$-covered on foliations of 3-manifolds

We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be $\mathbb{R}$-covered. This condition can be readily verified for many examples. Further, if an $\mathbb{R}$-covered foliation has a compact leaf $L$, then any transverse loop meeting $L$ lifts to a copy of the leaf space, and the ambient manifold fibers over $S^1$ with $L$ as fiber.