The large-canopy dimension conjecture for alternating-move games

Let $T\subseteq \mathbb{Y}$ be a subtree, and write

d=\dim_{\mathcal{H}}\left(\left\llbracket T\right\rrbracket\right).

For $d>\frac{1}{2}$ and $\delta\in\left[\frac{1}{2},d\right]$, consider win-lose alternating-move games on $T$ with payoff sets contained in \left\llbracket T\right\rrbracket. Large-canopy dimension conjecture. There exist sets W,W^I,W^{II}\subseteq\left\llbracket T\right\rrbracket such that

$$\dim_{\mathcal{H}}(W)=\dim_{\mathcal{H}}(W^I)=\dim_{\mathcal{H}}(W^{II})=\delta,$$

where $W$ is non-determined, Player I can guarantee a win in the game $G_I=\left(T,W^I\right)$, and Player II can guarantee a win in the game $G_{II}=\left(T,W^{II}\right)$. This would extend the analogous result for the full tree to subtrees whose canopies have Hausdorff dimension greater than $\frac{1}{2}$, while the preceding examples show that the conclusion can fail when the canopy has dimension at most $\frac{1}{2}$.