Memory can be deceiving. I think we are both right and both wrong..

The quote about "Chicago being responsible to New York" was actually not from Lefty but from his capo Mike Sabella in the context of the arbitration of the aborted business deal between Conte and Frankie Bals.

Bals was not allowed by Chicago to enter into independent agreements with supposed NY mobsters. The evident compromise worked out between NY and Chicago would have been that since Frankie Bals agreed to the deal , if Conte had been a real mobster he would have been allowed to operate his business in Milwaukee but without any hindrance from the Milwaukee mobsters.

Given that Chicago nixed the deal it would seem to indicate that the Outfit remained independent.

Lefty also said that Kansas City, Cleveland and Detroit belonged to Frank. When Lefty was speaking of the Milwaukee Mob he said they were crazy and couldn't be controlled by NY or anyone else, only Chicago.

So I think that NY and Chicago were roughly equivalent in terms of authority. I certainly don't see Aiuppa or Accardo "answering" to NY.