Optimal-strategy limit distribution conjecture for semi-restricted Rock, Paper, Scissors

Consider the semi-restricted Rock, Paper, Scissors game with score $S_n$, and let $S_n^{\mathrm{op}}$ denote the score when both players use optimal strategies. Let $W$ and $Z_1,Z_2,Z_3$ be the random variables from Theorem~, and define

$$Z_{\max}:=\max\{Z_1,Z_2,Z_3\}.$$

The variables $W$ and $Z_{\max}$ may be dependent. Optimal-strategy limit distribution conjecture. If both players play optimally, then

$$n^{-1/2} S_n^{\mathrm{op}}\overset{\mathrm{d}}{\longrightarrow} \mathcal S^{\mathrm{op}}=W+Z_{\max}.$$

The preceding theorem motivates this conjecture, but the supplied text describes the optimal strategy for one player as unknown and provides no resolution of the conjecture.