User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191031011444

I think that it is good to summarise the situations: It is because FGH is not directly related to proof theory. I guess that if either one of them occurs for well-known fundamental sequences, then it should be the latter one, because the breaking of FGH has been observed only when we created a pathologic example on purpose. (Of course, it is not an evidence. We do not have an effective general method to estimate FGH if it is not restricted to specific pathologic examples.)
 * 1) The FGH can break. Namely, \(f_{\textrm{PTO}(T)}\) does not necessary surpass all computable functions provably total under \(T\).
 * 2) The FGH perhaps can be too strong. Namely, \(f_{\textrm{PTO}(T)}\) might be un-comparably greater than the proof-theoretic diagonalisation (\(X\) in the blog post) of all computable functions provably to total under \(T\).