The strongly star-Hurewicz game nonimplication

Let $X$ be a topological space. The strongly star-Hurewicz property means that for every sequence $(\mathcal U_n:n\in\mathbb N)$ of open covers of $X$, there are finite sets $A_n\subseteq X$ such that $\{\operatorname{St}(A_n,\mathcal U_n):n\in\mathbb N\}$ is a $\gamma$-cover of $X$. In the strongly star-Hurewicz game, ONE chooses an open cover $\mathcal U_n$ in inning $n$, and TWO responds with a finite set $A_n\subseteq X$; TWO wins when $\{\operatorname{St}(A_n,\mathcal U_n):n\in\mathbb N\}$ is a $\gamma$-cover of $X$. The strongly star-Hurewicz game conjecture. The strongly star-Hurewicz property of $X$ need not imply that ONE does not have a winning strategy in the strongly star-Hurewicz game played on $X$. The statement concerns the possible failure of the converse to the evident implication from the absence of a winning strategy for ONE to the strongly star-Hurewicz property; the source provides no resolution or example.