Hyperspaces of arcs and two-point sets in dendroids

Let X be a metric continuum. Let A(X)={A⊂X: A is an arc or a one-point set} and F2(X)={A⊂X: A is nonempty and A has at most two points}. In this paper we show that if X is a fan (i.e., X is a dendroid with exactly one ramification point) and A(X) is homeomorphic to F2(X), then X is a dendrite. The question whether this implication is true for all dendroids is still open.