Wellfoundedness proof with the maximal distinguished set

In arXiv:2208.12944 it is shown that an ordinal ${\sup }_{N<\omega }{\psi }_{{\mathrm{\Omega }}_1}({\epsilon }_{{\mathrm{\Omega }}_{\mathbb{S}+N}+1})$ is an upper bound for the proof-theoretic ordinal of a set theory $\mathsf{K}\mathsf{P}{\mathrm{\ell }}^r+(M{\prec }_{{\mathrm{\Sigma }}_1}V)$. In this paper we show that a second order arithmetic ${\mathrm{\Sigma }}_2^{1-}-CA+{\mathrm{\Pi }}_1^1-C{A}_0$ proves the wellfoundedness up to ${\psi }_{{\mathrm{\Omega }}_1}({\epsilon }_{{\mathrm{\Omega }}_{\mathbb{S}+N+1}})$ for each $N$. It is easy to interpret ${\mathrm{\Sigma }}_2^{1-}-CA+{\mathrm{\Pi }}_1^1-C{A}_0$ in $\mathsf{K}\mathsf{P}{\mathrm{\ell }}^r+(M{\prec }_{{\mathrm{\Sigma }}_1}V)$.