The winning criterion for the ternary badly approximable set

Let $\alpha,\beta$ be Schmidt-game parameters, and let $\gamma_A$ and $\gamma_B$ denote the auxiliary constants associated with Alice's and Bob's moves. Let $\widetilde{\operatorname{3-BA}}(1/6)$ denote the modified ternary badly approximable set with parameter $1/6$. The winning criterion. If

$$\beta\gamma_A \geq \gamma_B,$$

then the set $\widetilde{\operatorname{3-BA}}(1/6)$ is $(\alpha,\beta)$-winning; if not, then it is $(\alpha,\beta)$-losing. This gives a sharp dichotomy for the stated ternary Schmidt-game set according to the comparison of the auxiliary move parameters. The supplied text does not state any resolution beyond the assertion itself.