On $C^0$-stability of compact leaves with amenable fundamental group

In his work on the generalization of the Reeb stability theorem, Thurston conjectured that if the fundamental group of a compact leaf $L$ in a codimension-one transversely orientable foliation is amenable and if the first cohomology group $H^1(L;\mathbb{R})$ is trivial, then $L$ has a neighborhood foliated as a product. This was later proved as a consequence of Witte-Morris' theorem on the local indicability of amenable left orderable groups and Navas' theorem on the left orderability of the group of germs of orientation-preserving homeomorphisms of the real line at the origin. In this note, we prove that Thurston's conjecture also holds for any foliation that is sufficiently close to the original foliation. Hence, if the fundamental group $\pi_1(L)$ is amenable and $H^1(L;\mathbb{R})=0$, then for every transversely orientable codimension-one foliation $\mathcal{F}$ having $L$ as a leaf, there is a neighborhood of $\mathcal{F}$ in the space of $C^{1,0}$ foliations with Epstein $C^0$ topology consisting entirely of foliations that are locally a product $L \times \mathbb{R}$.