Fan recognition from common-vertex arc decompositions in 1-dimensional continua

Determine whether every 1-dimensional continuum X that can be written as the union of a family of arcs {L} whose pairwise intersections are exactly a single common point v (that is, there exists v in X and a collection of arcs L with X = ⋃L and L ∩ L' = {v} for all distinct L, L' in L) must necessarily satisfy that the intersection of any two subcontinua of X is connected; equivalently, determine whether every such X is a fan.

Background

A fan is defined as a continuum X for which there exists a point v and a collection of arcs L with X equal to the union of L and pairwise intersections L ∩ L' = {v}, and such that the intersection of any two subcontinua of X is connected. Fans are known to be 1-dimensional, and the vertex v is unique.

The cited open question asks whether, in the 1-dimensional setting, the common-vertex arc decomposition alone forces the connected intersection property of subcontinua, i.e., whether condition (1) implies condition (2) and thus X is a fan. Prior work indicates the question has a positive answer under additional hypotheses, including the Jure property, but the general case remains open.