Hofbauer's converse conjecture for stable dynamics in bimatrix games

Let $(A,B)$ be a bimatrix game with a unique Nash equilibrium point in the interior $\mathring\Sigma$ of the strategy space $\Sigma$. Suppose this equilibrium is stable under either the best-response (BR) dynamics or the fictitious-play (FP) dynamics. Hofbauer's converse conjecture. The game $(A,B)$ must be a zero-sum game, meaning that it is linearly equivalent to a game $(C,D)$ satisfying $C+D=0$. This is the converse of the convergence theorem for zero-sum games: zero-sum games have orbits converging to the set of Nash equilibria, whereas it remains open whether the stated stability and uniqueness properties force a game to be zero-sum.