Hinkkanen–Martin conjecture on completely invariant Julia sets of rational semigroups

Let $G$ be a rational semigroup containing maps $f$ and $g$ with $J_f\neq J_g$, and let $E(G)$ denote its completely invariant Julia set. Assume that $E(G)$ is not the whole Riemann sphere.

Hinkkanen–Martin's conjecture. The set $E(G)$ is Möbius equivalent to a line segment or a circle.

The earlier conjectures that $E(G)$ must be a simple closed curve, or that the complement $W(G)$ has exactly two simply connected components bounded by $E(G)$, are refuted by examples in the paper. The modified statement is presented as currently unresolved; known examples have completely invariant Julia sets Möbius equivalent to a line segment or a circle.