Let $M$ be a compact smooth manifold, and $\mathcal{F}$ be a foliation on $M$. Assume that $L$ is a leaf of $\mathcal{F}$ for which there is $x\in L$ with the property that every neighborhood of $x$ in $M$ intersects a compact leaf of $\mathcal{F}$. Must $L$ itself be compact?

Note that the existence of such $x$ implies that all points of $L$ have the same property.

The conclusion that $L$ is compact seems to be true when the codimension of $\mathcal{F}$ equals one, and this goes back to Reeb. I am interested in what happens in general codimension, but particularly in the case where the codimension of $\mathcal{F}$ equals two.

Thanks.