User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-30118230-20180129215106

So far there is no proof backing up either of our claims,so I will just say that this is completely opinion-based,but I personally believe $$C(C(\Omega_2\omega,0),0)$$ reaches second order arithmetic. I am working on a proof for that,so we'll see ;)

There are several proposed main possibilities on the strength of TON:

(1) Weaker than Z2;

(2) As strong as Z2;

(3) As strong as Rudimentary set theory + "there exists an n-subtle cardinal"  for n is a natural number ;

(4) As strong as Z2 + PD;

(5) Somewhere in between those above;

I guess we just disagree on this,since I am one of the most optimistic people when it comes to the strength of TON.