The three-color pseudo-disk arrangement conjecture

Let $\mathcal A$ be a pseudo-disk arrangement, meaning a family of planar bodies whose boundaries are Jordan curves and such that the boundary of any member intersects the boundary of any other member in a connected curve. Let $S$ be a finite set of points. The pseudo-disk arrangement conjecture. Any finite set of points admits a $3$-coloring such that every pseudo-disk in the arrangement containing at least $3$ points contains two points with different colors. The source presents this as a stronger conjecture than the convex-set homothetic-copy statement and gives no resolution.