Is there a (known) analog to the Kunen inconsistency for large universe axioms in type theory?

Question

Over ZFC, the Kunen inconsistency gives a bound on how strong large cardinal properties can be. Moreover, this bound at the moment at least seems to be rather sharp, as Paul Carozza's Wholeness Axiom (WA) is basically a small modification of the Kunen inconsistency, and has yet to be shown inconsistent with ZFC.

In his paper "Hilbert Brouwer Controversy Resolved?", Per Martin Lof discusses the analogous notion of large universe axioms over MLTT, and mentions that MLM = MLTT + a mahlo universe seems to be the largest extension of MLTT that maintains constructive justification (in Martin Lof's sense) that we cannot progress non-trivially beyond without stepping into impredicativity.

The approach in (classical set theory) seems to be roughly:

How can we naturally progress to more and more expressive extensions of ZFC without reaching an inconsistency?

Whereas analogously, the approach with large universe axioms is more like:

How can we naturally progress to more and more expressive extensions of MLTT without reaching something not constructively justifiable?

Thus, keeping this analogy in mind, it seems to me that MLM is somewhat analogous to WA, which leads to the question: Is there a known large universe axiom which seems to bound "constructive jutifiability" for extensions of MLTT in the same way that the Kunen inconsistency bounds consistency for extensions of ZFC?

Answer 1

Yes, Michael Rathjen outlines such a result in his paper "The constructive Hilbert program and the limits of Martin-Löf type theory".

From what I can tell Rathjen's argument shows that the proof ordinal of $\Pi^1_2-CA_0$ is an upper bound on how much proof theoretic power an extension of MLTT can obtain without stepping into impredicativiy, but of course this might not be the sharpest possible bound.