Horizon-map determination of the set of minima

For a set of minima, let $C_{\mathrm{max}}$ be the largest set of curves containing $C$ for which every nonfilling subset $C'$ of $C_{\mathrm{max}}$ gives $\mathrm{Min}(C')^{\infty}$ adherent to $\mathrm{Min}(C)$. The set $C_{\mathrm{max}}$ need not have the property that its curves intersect pairwise at most once. Horizon-map determination conjecture. The image of the horizon map uniquely determines the set of minima

$$\mathrm{Min}(C)=\mathrm{Min}(C_{\mathrm{max}}).$$

Intuitively, a set of minima behaves like a convex hull of the minima on its boundary, and the horizon map determines this boundary. The supplied text does not state whether the conjecture has been resolved.