"$user" in a search_path string is replaced by CURRENT_USER not
SESSION_USER.  (It actually was SESSION_USER in the initial implementation,
but we changed it shortly later, and evidently forgot to fix the docs to
match.)

Noted by antonov@stdpr.ru

Discussion: https://postgr.es/m/159151fb45d490c8d31ea9707e9ba99d@stdpr.ru