I'm interested in this question:

Is there a star Lindelöf topological group which is not star countable?

A topological space $X$ is said to be star countable if whenever $\mathscr{U}$ is an open cover of $X$, there is a countable subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.

A topological space $X$ is said to be star Lindelöf if whenever $\mathscr{U}$ is an open cover of $X$, there is a Lindelöf subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.

Note that every star countable space is star Lindelöf, and a star Lindelöf space may not be star countable.

Since a Lindelöf topological group may not be countable or separable, thus I believe a star Lindelöf topological group may not be star countable. But I can not find a counterexample.

Could you help me? Thanks ahead.