A conjecture about the efficiency of first price mechanisms

We present different versions of a conjecture which would express that first price mechanisms never work very badly in a very general class of problems. The definitions include most of the problems where there is a principal (seller) who has the right to exclude others from the game. The exact definitions are motivated by the "first price mechanism" in E Cs: "Efficient Teamwork", but the conjecture is relevant for most auction problems, e.g. for combinatorial auctions.