User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180201150715/@comment-30118230-20180201164832

Like I said,TON is the only notation strong enough to reach the limit of TON. I don't have a proof for it,but I'm pretty damn sure of it. And I don't believe such a simple notation like BMS can really be as strong as TON. I would be surprised if it even reaches $$C(\Omega_2 2+C(\Omega_2 +C(\Omega_2,C(\Omega_2,C(\Omega_2 2,0))),0),0)$$.

I also agree with KobLev's comment completely.