Star compact topological groups

Question

Recently I'm interested in this open question:

Must every star compact topological group be countably compact?

1. star compactness ( which implies pseudocompactness ) = for every open cover $U$ of the space $X$, there exists a compact subspace $K$ such that $\cup \{u \in U: u \cap K \text{ is not empty} \} = X.$
2. countably compact ( which implies star compactness obviously) = for every open cover $U$ of the space $X$, there exists a finite subspace $K$ such that $\cup \{u \in U: u \cap K \text{ is not empty} \} = X.$ This definition is under the $T_1$ assumed. It is equivalent to this: for every countable open cover of $X$ there is a finite subcover of $X$.

I'm not very familar with topological group. I have some questions:

Firstly, could someone complete to list the properties of star compact topological group. These of star compact topological group are I know, for example:

1. Tychonoff = $T_0$ in every topological group,
2. pseudocompactness from star compactness,
3. CCC = countable chain condition, for every pseudocompact topological group has the CCC.

Secondly, if you have any idea for this open question, you could write here.

Thirdly, Is there a pseudocompact topological group but is not separable?

Thanks for any help:)