I haven’t even been able to identify a programme from Sony by which roots could apply to be added to the trusted list. So basically some unknown set of roots is trusted, those work, others get the error. Nice. One more reason to avoid using your PS4 for web browsing I guess, but it does mean we can’t even begin to fix this.
There’s a fair chance it’s using (some old version of) the Mozilla NSS root list, as shipped with many Free Software operating systems and tools (and of course Firefox itself). A PS4 owner could verify that either by painstakingly testing sites which use certs from particular CAs or maybe it’s documented in the fine print somewhere.