This “chairman” position is more of a media term. It’s not an actual position as each boss is “equal” on paper. But as we know there’s always hidden alliances and interests so unofficially there’s usually one of the bosses that during certain periods holds the most power, like Gigante in the mid 80s…meaning the other bosses wouldn’t say Gigante is the chairman but they’d all understand (whether they admit it or not) he’s the most influential and powerful.

I’m also not sure that big Paul was ever more powerful than the genovese boss. The Genovese forced him to kill a Gambino capo even. I’m pretty sure that since Carlo passed that the Genovese were more powerful. And it’s not been ever really that close, especially 12/16/85 - present.