Strong convergence conjecture for imitation dynamics

Let $\mathbf{A}$ be a game matrix and let $\alpha$ be a learning rate. Let $I_G(t)$ denote the relevant maximal-set quantity in the imitation dynamics, and let $\alpha^*$ be a learning-rate bound. Strong convergence conjecture. For every game matrix $\mathbf{A}$, there exists a learning rate $\alpha^*$ such that the imitation dynamics always converge when $\alpha\leq\alpha^*$. The source presents this as the second of two general convergence conjectures and leaves it as an open problem; no proof or resolution is given.