The auto-clonvex characterization conjecture

Let $S\subseteq\mathbb C$. Call $S$ auto-clonvex when it is $S$-clonvex, meaning that it is closed and $\lambda$-convex for every $\lambda\in S$. For each parameter $\lambda$, the set $R_\lambda$ is known to be auto-clonvex. The auto-clonvex characterization conjecture. If $S$ contains $0$ and $1$ and is auto-clonvex, then there exists $\lambda\in\mathbb C$ such that

$$S=R_\lambda.$$

The authors state this as the conjectured converse to the known auto-clonvexity result; no resolution is given.