Antimagicness of disjoint unions of join graphs

Determine which disjoint unions of join graphs are antimagic, including those disconnected cographs that can be represented as such unions.

Background

The paper proves that every join graph other than K_2 is antimagic. It notes that complements of disconnected graphs are join graphs, but the antimagicness of disjoint unions of join graphs is not resolved by the main theorem. This class includes disconnected cographs, making its study a broader extension of the results established in the paper.