$\Omega$: 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 $n$.

A space $X$ is said to be Menger (respectively, Scheepers) if $X$ has the Menger (respectively, Scheepers) property. It is well known that every Scheepers space is Menger but the coverse is not true (see here). We are looking for a ZFC example of a Menger space which is not Scheepers (here, it means without any additional set-theoretic assumption, is there a Menger space which is not Scheepers?).