McNulty's fundamental-triangle conjecture for $R_\lambda$

For a parameter $\lambda\in\mathbb C$, call the triangle with vertices $0$, $1$, and $\lambda$ the fundamental triangle. Let $R_\lambda$ be the associated closure. McNulty's fundamental-triangle conjecture. If $R_\lambda$ contains a point in the interior of the fundamental triangle, then $R_\lambda$ is convex. This conjecture is attributed to George McNulty and is presented without a resolution.