From the third comment on archive.org : "The three novels which make up the original Foundation trilogy were written as a magazine serial between 1940 and 1950, for a science fiction magazine edited by the legendary John W Campbell. They were thus composed as a series of short stories, giving them a number of advantages over the later books by which Asimov added to the story forty years afterwards."

So there is no Asimov copyright problem as it's based on Campbell's work.

Yet it's presented as Assimov's trilogy. So I don't know what to believe.