Chicago is right. The Chicago boss had a seat on the Commission just like the New York bosses and were equal. Its connection to the Genovese Family went way back to the connection between Capone and Joe Masseria. The Genovese Family had an emissary to Chicago because of their closeness: Gaetano Ricci (AKA Tony Gobbels). In the 1960s the Outfit became less involved with Commission because it was almost exclusively focused on New York issues. Chicago was big enough that it didn't need New York.

Maybe the poster Chicago can answer questions about how the Outfit related to New York after the 1960s, but to me it seems that it focused on the Midwest and the West and had no interest in East Coast matters.