Metric convex-hull equality for transition attractors

Let $({\mathbb X},d)$ be a metric space, and call $[x,y]:=\{z\in{\mathbb X}:d(x,z)+d(z,y)=d(x,y)\}$ the segment with ends $x,y$. A set is metrically convex if it contains $[x,y]$ for every pair of its points, and its metrically convex hull is $\operatorname{conv}_d S:=\bigcup_{x,y\in S}[x,y]$. Let $F_t$ be a family of functions that maps metrically convex sets onto metrically convex sets. The metric convex-hull conjecture. The metrically convex hulls of the lower and upper transition attractors coincide:

$$\operatorname{conv}_d A_{\bullet}=\operatorname{conv}_d A^{\bullet}.$$

This conjecture proposes that the two transition attractors have the same metrically convex closure under the stated invariance assumption. The supplied context does not state any partial results or resolution, so its status remains open.