Hector's conjecture on exceptional minimal sets

Let a finitely generated group of $C^2$ circle diffeomorphisms admit an exceptional minimal set $\Lambda$, meaning a minimal closed invariant subset of the circle that is homeomorphic to the Cantor set. Hector's conjecture. The Lebesgue measure of $\Lambda$ is zero. This conjecture concerns exceptional minimal sets for nonminimal group actions without finite orbits and is fundamental to the study of such actions; it was stated by G. Hector in 1977–78, and its general status is not resolved in the source.