User blog comment:Boboris02/Ordinal Analysis of Theories/@comment-11227630-20180213093537/@comment-30118230-20180213113323

Hmmm....I think you are right. It may be that $$\text{ACA}_0+\Pi^1_{n+1}-\text{BI}$$ has a PTO of $$\psi(\delta_n)$$ rather than $$\text{ACA}_0+\Pi^1_n-\text{BI}$$. I am looking for a source for it now.