Average-payoff dominance conjecture for fictitious play

Consider bimatrix games with a unique, completely mixed Nash equilibrium and their typical fictitious-play orbits. Average-payoff dominance conjecture. For most such games, typical fictitious-play orbits dominate the Nash equilibrium in terms of average payoff. In particular, this should hold under certain assumptions on the game's transition combinatorics, such as when each pure strategy invokes a distinct pure best response. This is presented as a strengthening of the preceding conjecture: it predicts average-payoff improvement along typical fictitious-play behavior for most games, but the supplied text provides no proof or resolution.