Cantor set of arcs in Bellamy dendroids

Determine whether every Bellamy dendroid—that is, every smooth plane dendroid whose endpoint set E(X) is connected—necessarily contains a Cantor set of arcs, meaning a continuous family of pairwise-disjoint arcs whose decomposition space is a Cantor set.

Background

The paper studies structural properties of endpoints of smooth plane dendroids and introduces the term Bellamy dendroid for those with connected endpoint sets E(X). A central theme is relating endpoint structure to the presence of large families of disjoint arcs (Cantor sets of arcs), which correspond to non-Suslinian behavior.

The authors prove that if the endpoint set E(X) is 1-dimensional, then the dendroid contains either a Bellamy dendroid or a Cantor set of arcs, and they further analyze when hereditarily disconnected endpoint sets force non-Suslinian behavior. Against this backdrop, the question asks whether connected endpoint sets are strong enough to force the presence of a Cantor set of arcs in all Bellamy dendroids.