Given a Tychonoff space $X$, let $\Omega$ be the set of all $\omega$-coverings of $X$, where by $\omega$-cover we mean a collection $\mathcal{U}$ of open sets of $X$ such that for any $F\in[X]^{<\omega}$ there exists $U\in\mathcal{U}$ such that $F\subset U$.
Now, let G$_1(\Omega,\Omega)$ be the game between Player I and Player II defined as follows: for every inning $n<\omega$, the Player I chooses an $\omega$-cover $\mathcal{U}_n$, and then Player II picks $U_n\in\mathcal{U}_n$; Player II wins iff $\{U_n:n\in\omega\}\in\Omega$.
Clearly, if the Player II has a winning strategy in the game $\mathrm{G}_1(\Omega,\Omega)$, then the Player I does not have a winning strategy in the same game.
I would like to know any references with examples of spaces where the game $\mathrm{G}_1(\Omega,\Omega)$ is undetermined, i.e., such that both players do not have winning strategies in the game $\mathrm{G}_1(\Omega,\Omega)$.