Does there exist a perfect and star Lindelöf space which is not separable?

Question

Does there exist a perfect and star Lindelöf space which is not separable?

A topological space in which every closed set is a $G_\delta$-set is called a perfect space.

A topological space $X$ is said to be star Lindelöf, if for any open cover $\mathcal U$ of $X$ there is a Lindelöf subspace $A \subset X$ such that $\operatorname{St}(A, \mathcal U)=X$.

Thanks for your help.

Answer 1

There are certainly counterexamples: all $L$-spaces will do (an $L$-space is a hereditarily Lindelöf regular space, which is not separable), Such spaces are perfect (because open sets are $F_\sigma$, and so closed sets are $G_\delta$), and trivially star Lindelöf (as any Lindelöf space is).

Moore constructed an $L$-space in ZFC (I now recall).