Conjectured extension for unions of paths of length two

Establish that the disjoint union kP_3 of k paths on three vertices is r-EKR for every admissible integer r, not only for r ≤ μ(kP_3)/2.

Background

The survey reports that the case kP_3 was proved for all r ≤ μ(kP_3)/2 = k/2. It then records the stronger conjecture that the same graph satisfies the star-maximality property for every r.

The difficulty identified is that kP_3 is not vertex-transitive: its middle vertices and leaves form distinct vertex orbits, complicating the uniform counting arguments used for disjoint unions of edges.