Minimal smooth group actions on the circle are Lebesgue-ergodic

Let a finitely generated group act smoothly and minimally on the circle, meaning that every orbit is dense. Let Lebesgue measure denote the standard Lebesgue measure on the circle. Minimal-action ergodicity conjecture. Every minimal smooth action of a finitely generated group on the circle is ergodic with respect to the Lebesgue measure. The conjecture is the one-dimensional case of the question of when a smooth action on a compact manifold must be Lebesgue-ergodic; the paper proves it under additional assumptions, while the stated general form is not resolved here.