Weak convergence conjecture for imitation dynamics

Let $\mathbf{A}$ be a game matrix and let $\alpha$ be a learning rate. Suppose that the maximal set of $I_G(t)$ is constant for all $t\geq t_0$. Let $\mathcal{X}^*$ denote the maximal set and let $\mathcal{H}(\mathcal{X}^*)$ denote its convex hull. Weak convergence conjecture. There exists a rate $\alpha^*$ such that, if $\alpha<\alpha^*$, the dynamics have an attracting fixed point of the imitation dynamics contained in $\mathcal{H}(\mathcal{X}^*)$, and the dynamics converge to that fixed point. The conjecture concerns general convergence of imitation dynamics, which are not strict contractions; the source presents it as an open problem and gives examples of convergence but no proof of the claim.