Regularity conjecture for quasiconformal homeomorphisms of rigid Carnot groups

Let $G$ be a Carnot group. A group is rigid if the space of smooth contact embeddings $G\supset U\to G$ is finite-dimensional for every connected open subset $U\subset G$. Regularity conjecture. If $G$ is a rigid Carnot group, then every quasiconformal homeomorphism

$$G\supset U\to U'\subset G$$

is $C^\infty$. This conjecture proposes that quasiconformal self-maps of rigid Carnot groups have the same smooth regularity as contact diffeomorphisms; whether it holds in full generality is not established in the supplied source.