This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
2026-04-01 12:46:38.1775047598
Is every regular paratopological group completely regular?
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Yes, this problem is still open for many years and seems to be very hard. If you need a reference for this, you can add it, for instance, as "[Rav] Alex Ravsky. A personal communication". :-)