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The following bug has been logged on the website:

Bug reference:      16592
Logged by:          Barry Pederson
Email address:      bp@barryp.org
PostgreSQL version: 12.4
Operating system:   Ubuntu 20.04
Description:        

In the regression tests src/test/regress/expected/tsearch.out there's an
example relating to double-quoted searches:

select websearch_to_tsquery('english', '"pg_class pg"');
    websearch_to_tsquery     
-----------------------------
 ( 'pg' & 'class' ) <-> 'pg'
(1 row)

However, if you ran a query like that against that exact text 'pg_class pg',
the result is false when I'd expect it should be true

   select to_tsvector('pg_class pg') @@ websearch_to_tsquery('"pg_class
pg"');

I'd think websearch_to_tsquery('english', '"pg_class pg"') should ideally
return
  
    'pg' <-> 'class'  <-> 'pg'

which does match the original string, e.g:

      select to_tsvector('pg_class pg') @@ to_tsquery('pg <-> class <->
pg');

does return true