A regular space $X$ is

- star compact (which implies pseudocompact)
- with $G_\delta$-diagonal
- star countable
- first countable
- $e(X)\le \aleph_0$ ( in fact it implies star countable)
- $|X|=\aleph_1$
- Cech-complete
- under CH

My question is this: Must $X$ be countably compact?

Thanks ahead.