The Banaschewski-Lawson-Ershov observation on separate vs. joint continuity

Joint continuity is a stronger property than separate continuity. In what cases are those properties equivalent? The question was solved, partially, by Yuri Ershov in 1997, and completely by Bernhard Banaschewski in 1977 (apparently with a gap in the proof) and by Jimmie Lawson in 1985. The answer has to do with locally finitary compact spaces: read the full post.