Recently, Taranovsky has updated his ordinal notation page. In the last section he defined another ordinal notation system. In that system he used a ternary function C, and \(C_i(a,b)\) corresponds to \(C(\Omega_2\times i+a,b)\) in the main system.

Taranovsky conjectured that using \(i\le n\) (where n is a positive integer), the last system reaches \(\Pi_n^1\text{-TR}_0\). As a result, in the main system, \(C(C(\Omega_2\omega,0),0)\) reaches second-order arithmetic.

I've done some comparisons between array notation and Taranovsky's ordinal notation (but only a part of them published on my site). If the conjecture and the comparisons work, s(n,n{1{1(2{2+}2)+2}2(2{2+}2)+2}2) will surpass all functions provably recursive in second-order arithmetic.