"as the majors backed out of exploration activities"... that`s the catch Donker. They are supposed to perform on their leases..just like you were. They are harvesting and don`t want competition. The only way would be to break up the joint-operating unit agreement between the three of them that gives them all a veto over anything either of the other two contemplate. 
If Exxon doesn`t want to drill on a lease because it would cost them, they don`t drill, even if it would benefit BP or Conoco.  
That certainly blocks serious entrants that could bring in "close" oil that isn`t 150 miles from TAPS facilities. The state can`t afford to subsidize your or Caelus` connector lines Donk. Sorry.