$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $U\in\mathcal U$ such that $F\subseteq U$.

A space $X$ is said to have the Menger property if for each sequence $(\mathcal U_n)$ of open covers of $X$ there exists a sequence $(\mathcal V_n)$ such that for each $n$ $\mathcal V_n$ is a finite subset of $\mathcal U_n$ and $\{\cup\mathcal V_n : n\in\mathbb N\}$ covers $X$.

A space $X$ is said to have the Scheepers property if for each sequence $(\mathcal U_n)$ of open covers of $X$ there exists a sequence $(\mathcal V_n)$ such that for each $n$ $\mathcal V_n$ is a finite subset of $\mathcal U_n$ and $\{\cup\mathcal V_n : n\in\mathbb N\}\in\Omega$ or $\cup\mathcal V_n=X$ for some $n$.

The infinitely long games correponding to the Menger and Scheepers properties are defined as follows.

The Menger game on $X$ is played as follows. Players ONE and TWO play an inning for each positive integer $n$. In the $n$th inning ONE chooses an open cover $\mathcal U_n$ of $X$ and TWO responds by selecting a finite subset $\mathcal V_n$ of $\mathcal U_n$. TWO wins the play $\mathcal U_1,\mathcal V_1,\dotsc,\mathcal U_n,\mathcal V_n,\dotsc$ if $\{\cup\mathcal V_n : n\in\mathbb N\}$ covers $X$; otherwise ONE wins.

The Scheepers game on $X$ is played as follows. Players ONE and TWO play an inning for each positive integer $n$. In the $n$th inning ONE chooses an open cover $\mathcal U_n$ of $X$ and TWO responds by selecting a finite subset $\mathcal V_n$ of $\mathcal U_n$. TWO wins the play $\mathcal U_1,\mathcal V_1,\dotsc,\mathcal U_n,\mathcal V_n,\dotsc$ if $\{\cup\mathcal V_n : n\in\mathbb N\}\in\Omega$ or $\cup\mathcal V_n=X$ for some $n$; otherwise ONE wins.

It is well known that a space $X$ has the Menger property if and only if ONE does not have a winning strategy in the Menger game on $X$ (see here). We try to prove a similar result for the Scheepers game, but we fail. Does a similar result hold for the Scheepers game?