To quote the manual at https://www.php.net/manual/en/features.persistent-connections.php:
'When a persistent connection is requested, PHP checks if there's already an identical persistent connection (that remained open from earlier) - and if it exists, it uses it. If it does not exist, it creates the link. An 'identical' connection is a connection that was opened to the same host, with the same username and the same password (where applicable).'
That seems to imply that there no need to check prior to trying to connect.
Note that the article goes into a lot of considerations about how persistent connections work out with web servers, particularly about how subsequent processes attempting to use the connection may be blocked by what a fault in an earlier process created, implying that persistent connections may not be reliable in a web server environment given how web sessions can terminate at any time.
In particular, it recommends that persistent connections are not used for scripts that use table locks or transactions.