User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-11227630-20180203000827/@comment-1605058-20180203112715

The answer is no. ZFC proves the reflection theorem, which states that for any sentence \(\varphi\) true in \(V\), there is an ordinal \(\beta\) such that \(V_\beta\) satisfies \(\varphi\) and also satisfies ZFC with axiom schemata restricted to bounded complexity, in particular - KP holds. Taking \(\varphi\) to be "\(\alpha\) exists", we get that ZFC + "\(\alpha\) exists" imples KP  + "\(\alpha\) exists" has a standard model. For brevity let me call the two theories T1 and T2. I claim T1  proves the PTO of T2 well-ordered, which implies T1 has PTO larger tham T2.

Indeed, existence of the standard model implies that whenever T2 proves "this recursive order is a well-order", then the recursive order is indeed a well-order. Consider  the Turing machine which finds all proofs from T2 that some recursive order is a well-order, and then constructs an ordering which is a concatenation of those orderings. Then T1 easily proves this order is a well-order and its length is at least (actually equal to) the PTO of T2.