Equivalence between uniform spiders and unions of paths

Determine whether an ℓ-uniform spider with k leaves is r-EKR if and only if the disjoint union kP_{ℓ+1} of k paths on ℓ+1 vertices is r-EKR.

Background

Uniform spiders and disjoint unions of paths arise as related families in the survey’s study of trees and graph unions. The proposed equivalence would transfer the EKR property between a connected tree construction and a disconnected path construction.

The survey presents this as an explicit unresolved question alongside the broader problem of determining whether all spiders are r-EKR.