  In a zero-sum stochastic game with signals, at each stage, two adversary
players take decisions and receive a stage payoff determined by these decisions
and a variable called state. The state follows a Markov chain, that is
controlled by both players. Actions and states are imperfectly observed by
players, who receive a private signal at each stage. Mertens (ICM 1986)
conjectured two properties regarding games with long duration: first, that
limit value always exists, second, that when Player 1 is more informed than
Player 2, she can guarantee uniformly the limit value. These conjectures were
disproved recently by the author, but remain widely open in many subclasses. A
well-known particular subclass is the one of absorbing games with incomplete
information on both sides, in which the state can move at most once during the
game, and players get a private signal about it at the outset of the game. This
paper proves Mertens conjectures in this particular model, by introducing a new
approximation technique of belief dynamics, that is likely to generalize to
many other frameworks. In particular, this makes a significant step towards the
understanding of the following broad question: in which games do Mertens
conjectures hold?
