I'd say that the fact the Italians were involved and profiting from both Lucas and Barnes operations, makes it unlikely that they were threatened or rivaled by them. I'd also say that the five families were involved in very different rackets like Loansharking,Gambling,Extortion and Union related crimes, while neither Lucas or Barnes groups were this diverse or tried to impede on the Italians there. I think both decided to operate together and that the Italians profited much more from their crimes than vice-versa.

In all honesty though i'm not the person to ask, the knowledgabl,e posters on Black crime like BlackFamily or Scorcese, as well as all of the NY crime posters are.

Certainly not someone like cookcounty who professes to be an expert on any Chicago related crime even though he's usually wrong, and who has no problem chiming in on any other crime in the world without ever going further than his vague three sentence routine.