The strongly star-Scheepers game conjecture for closed-discrete and sigma-compact spaces

Let $X$ be a strongly star-Lindelöf space of the form $Y\cup Z$, where $Y$ is a closed discrete set and $Z$ is a $\sigma$-compact subset of $X$ with $Y\cap Z=\emptyset$. Let $\operatorname{SG_{fin}^\ast}(\mathcal{O},\Omega)$ be the game on $X$ in which ONE and TWO play according to the strongly star-Scheepers selection-game rules.

The strongly star-Scheepers game conjecture. If $X$ is strongly star-Scheepers, then ONE does not have a winning strategy in the game $\operatorname{SG_{fin}^\ast}(\mathcal{O},\Omega)$ on $X$.

This is proposed as a result related to the strongly star-Scheepers property, following analogous investigations for strongly star-Menger spaces. The supplied text does not indicate whether the claim has been proved or refuted.