A topological space has *calibre* $\aleph_1$ if for every uncountable sequence $\langle U_\alpha\mid\alpha\lt\aleph_1\rangle$ of nonempty open sets $U_\alpha\subset X$, there is an uncountable subfamily $\Lambda\subset\aleph_1$ with $\bigcap_{\alpha\in\Lambda}U_\alpha\neq\emptyset$.

Is there a calibre $\aleph_1$ Moore space which is not separable?

(Under CH, the answer is no, since every first countable calibre $\aleph_1$ space is separable.)

Thanks.