# Let $X$ be a Lindelöf space. Does the following condition imply that $X$ is star-Menger?

Let $$X$$ be a Lindelöf space. If every cover $$\{G_M : M\in\mathcal{M}(X)\}$$ of $$X$$, where $$G_M$$ is a $$G_\delta$$ subset of $$X$$ with $$M\subseteq G_M$$, has a subcover of cardinality $$\mathfrak{b}$$, then is $$X$$ star-Menger?

1. A space $$X$$ is said to be star-Menger if for every 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 $$\cup_{n\in\mathbb{N}}\{St(V,\mathcal{U}_n) : V\in\mathcal{V}_n\}$$ is an open cover of $$X$$.
2. $$\mathcal{M}(X)$$ is the collection of all star-Menger subspaces of $$X$$.
3. $$\mathfrak{b}$$ is the minimum cardinality of an unbounded subset of $$\mathbb{N}^\mathbb{N}$$.