A regular space $X$ is

1. star compact (which implies pseudocompact)
2. with $G_\delta$-diagonal
3. star countable
4. first countable
5. $e(X)\le \aleph_0$ ( in fact it implies star countable)
6. $|X|=\aleph_1$
7. Cech-complete

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

Thanks ahead.