Constant payoff in zero-sum stochastic games

In a zero-sum stochastic game, at each stage, two adversary players take decisions and receive a stage payoff determined by them and by a controlled random variable representing the state of nature. The total payoff is the normalized discounted sum of the stage payoffs. In this paper we solve the "constant payoff" conjecture formulated by Sorin, Venel and Vigeral (Sankhya A 72 (1) (2010) 237-245): if both players use optimal strategies, then for any alpha > 0, the expected discounted payoff between stage 1 and stage alpha/lambda tends to the limit discounted value of the game, as the discount rate lambda goes to 0.