In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this subject, can you comment?

**Propoition 7:** A $G$-space is compact if and only if it is simultaneously $G$-pseudocompact
(it is equivalent $\beta _{G}X=\beta X$, as mentioned in the article) and $G$-Hewitt.

[$G$-pseudocompact and $G$-Hewitt spaces](https://doi.org/10.1070/RM1980v035n06ABEH001975)