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I would like to know how is it going the research in compactifications of locally compact Hausdorff spaces. Are there people doing this? Are there relevant conjectures on it?

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I would not think it is particularly active, unless you are prepared to add more structure, for instance, study wonderful compactifications:

https://en.wikipedia.org/wiki/Wonderful_compactification

Having said that, I can recall a couple of open problems from Arkhangelski-Tkachenko's Topological Groups. I am not sure about their current status (the book is 2008). Let $X$ be Tychonoff space.

  1. When is the Cech-Stone compactification $c(X)$ a Moscow space?
  2. When does $X$ admit a Hausdorff Moscow compactification?
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  • $\begingroup$ Thank you. Actually, I already work with compactifications of spaces with more structure. I would like to know something that is purely topological like the two problems you've said. $\endgroup$
    – Lucas Henrique
    Commented Jul 15, 2019 at 16:08

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