Computational hardness of constructing Type III dynamics for games

Let $g$ be a normal-form game. A game is degenerate if it has a non-isolated set of Nash equilibria, and a dynamical system is of Type III as defined in the paper. Computational hardness conjecture. The computational task of finding, from the description of $g$, either a degeneracy of $g$ or an algorithm producing the direction of motion and speed of a dynamical system of Type III is PPAD-hard (and FIXP-hard for three or more players). This question lies at the boundary between computation, game theory, and the topology of dynamical systems; resolving it is expected to require new complexity-theoretic techniques for dynamical systems.