Faithful1, To answer your question. Chicago was allied with the very powerful Genovese Family. That's who they dealt with in New York. If anybody from Chicago went to New York on occasion, that's who they would see. I never heard about them seeing the Gambinos or anyone else.

It was understood that the relationship was between Chicago and The Genovese's and to be honest, both sides valued it. the Genovese's had an edge over the other 4 families having Chicago as their ally. Chicago had the same advantage having a powerhouse like the Genovese's as their ally.
During The Giancana reign which lasted about 8 years, Sometimes Mooney went to New York with Joey Glimco. Glimco was a strong made guy who was involved with the Unions and was direct with Mooney.

In the later years, Cerone would sometimes go to New York with Willie Messino, his driver & Bodyguard. Willie was very proud to go. I remember him saying to a couple of the other guys that he liked it that the Genovese top men saw that he was Jack's man.

The Genovese Family had a smaller type of partnership in some way with Chicago in Las Vegas that I never really understood. None of the other Families, only the Genovese's.

So, as you can see, there was no big competition really between New York and Chicago because Chicago was in with the Genovese Family. The Genovese Family liked Chicago better than they liked the Bonanno's. Far better.

The only completion I see is from a couple of delusional posters on this forum. LOL