I think it will be popular as it will be something different. Will probably a pre made deck decided by FFG that everyone plays against and see who does the best. Besides people play cooperative stuff all the time, I mean any team sport is both cooperative and competitive at the same time.
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I'm thinking more along the lines of (somehow) measuring each player's contribution to their game and have the top contributor(s) move on to team up with the top contributor(s) from other tables and keeping the cycle going. This way each table will police itself and each player will try his or her hardest to contribute. This should preserve the cooperative spirit of the game while still giving those who feel as though they need to prove they're better than everyone else a chance to do so.
That my friend is going to be insanely difficult - how are you going to measure top contribution?
Presumably by keeping track of each player's progress toward the overall goal (# of kills, # of resources gathered, # of quests completed, etc.)
I don't think it will be that complicated, it will probably just be a points system similar to Race for the Galaxy where you get one point value at the end of the game by adding up certain factors.
Not all who wander are lost...
maybe there is a way that an opposing player takes control over the Enemy Deck and has the cahnce to play this "deck" to a certain degree.
so for example a team consists of 4 players, 3 of them play the regular game in a round and the forth is reponsible to play the Enemy Deck of the opposing team (and vice versa).