Finite-limit-set conjecture for discounted stochastic-game values

Let $v_\lambda$ denote the value vector associated with the parameter $\lambda>0$ in the stochastic game, and let $V$ be a finite set. Finite-limit-set conjecture. There exists a finite set $V$ such that

$$\lim_{\lambda \to 0} v_\lambda\in V.$$

The preceding results establish existence of the limit of $v_\lambda$ and polynomial constraints on its behavior as $\lambda$ tends to zero; the conjecture asserts that the limiting value belongs to a finite, parameter-independent set.