Shot in the face, eh? Great way to cover up if it was really him.
DNA testing? everything Buff said + the fact that even if it wasn't him it is so good propaganda opportunity.

There is nearly 100% chance that he left instructions in case of his death(if its true), plus they are bound to have other iconic(and possibly intelligent) people to continue his 'work', also they now have a nice martyr and another reason to fight.