User blog comment:P進大好きbot/What is the greatest ordinal notation now?/@comment-35870936-20180623152451/@comment-1605058-20180624102745

Again, we have no proofs giving lower bounds on the strength of Taranovsky's notation. To the best of our knowledge, it could be weaker than whatever notation Rathjen has devised to study \(Pi^1_2-CA_0\); we literally don't know.

As for OCFs I will let someone else talk about them; I have never read Rathjen's (or others') work in detail and my understanding is very cursory.