Re: ('dog$house' = quote_ident('dog$house')) is surprisingly FALSE

The doc for "quote_ident()" says this:

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https://www.postgresql.org/docs/14/functions-string.html
Returns the given string suitably quoted to be used as an identifier in an SQL statement string. Quotes are added only if necessary (i.e., if the string contains non-identifier characters or would be case-folded). Embedded quotes are properly doubled.
»

But it's incorrect w.r.t. "quotes are added only if necessary" for this to return FALSE:

select 'dog$' = quote_ident('dog$');