Finally Boeing participated but the got exclusivity to this new engine in exchange.
Sounds the wrong way round to me. Boeing agreed to use only the GE engine on the 777LR (cf the CFM56 on the 737NG).
That's not the same as GE agreeing that only Boeing would be offered the engine (cf also offering the CFM56 on the A320).
Airbus is since jealously waiting for a similarly powerful engine
To power what, exactly ?