Winning-strategy conjecture for selective strong screenability games on cubes

For each positive integer $n$, let $\lbrack 0,1\rbrack^n$ denote the $n$-dimensional closed unit cube, and let $\mathcal{O}$ be the collection of open covers. In the game $\mathsf{G}_d^\alpha(\mathcal{O},\mathcal{O})$, played for length $\alpha$, ONE and TWO have the winning strategies specified below.

Winning-strategy conjecture. For each positive integer $n$, ONE has a winning strategy in

$$\mathsf{G}_d^{\omega\cdot n}(\mathcal{O},\mathcal{O}),$$

and TWO has a winning strategy in

$$\mathsf{G}_d^{\omega\cdot n+1}(\mathcal{O},\mathcal{O})$$

on \lbrack 0,1\r\brack^n.

These conclusions are suggested by the results for the closed unit interval and by heuristic arguments; the source does not state a resolution of the conjecture for all positive integers $n$.