Transitivity of arrows between impartial misère-game positions

Let $\alpha$, $\beta$, and $\gamma$ be impartial positions. An arrow $\delta\to\varepsilon$ exists when Left moving second can win $\varepsilon+\delta^{\circ}$, or under a similar criterion. Transitivity conjecture. If arrows $\alpha\to\beta$ and $\beta\to\gamma$ exist, then an arrow $\alpha\to\gamma$ exists. The conjecture would establish transitivity of the arrow relation on impartial positions and could help determine whether these positions form a Joyal-style category under a suitable sum and negative; the paper gives no resolution.