An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite.
A space $X$ is said to be star-$K$-Menger if for each sequence $(\mathcal U_n)$ of open covers of $X$ there exists a sequence $(K_n)$ of compact subsets of $X$ such that $\{St(K_n,\mathcal U_n) : n\in\mathbb N\}$ is an open cover of $X$.
A space $X$ is said to be star-$K$-Hurewicz if for each sequence $(\mathcal U_n)$ of open covers of $X$ there exists a sequence $(K_n)$ of compact subsets of $X$ such that $\{St(K_n,\mathcal U_n) : n\in\mathbb N\}$ is a $\gamma$-cover of $X$.
It clear that every star-$K$-Hurewicz space is star-$K$-Menger. We are looking for a ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz (here, it means without any additional set-theoretic assumption, is there a star-$K$-Menger space which is not star-$K$-Hurewicz?)