It is not very useful to add full text search to an email field used for login. A regular unique index, perhaps case insensitive, is what you should be using.
The positive response of the address verification will tell you the address is deliverable and the user has access to it. Later if someone tries to register a capitalized form of the address it'll get rejected because of that account ID collision. Then the user can be pushed to a password recovery path where they'll need access to the e-mail/MFA to get control of the account.
Amusingly, the context of this thread was in using case-insensitive search for email fields, but if emails are truly case sensitive, this is all moot, because you can only do direct comparisons.
If you run into a case where their e-mail server enforces case sensitivity they have bigger problems to deal with. E-mail has long been a system that requires loose adherence to the specs.
IIRC, as per spec/rfc, e-mail addresses ARE case sensitive.
However the de-facto standard is to ignore such thing and deliver emails for Bob@example.com, bob@example.com and BoB@example.com all to the same mailbox.
https://www.postgresql.org/docs/current/pgtrgm.html#id-1.11....