 Daniel Dzierzgowski has gone so far as to study an intuitionistic version of NF constructed in the same way; all that we can usefully report here is that it is not clear that the resulting theory INF is as strong as NF (in particular, it is unclear whether INF interprets Heyting Arithmetic, because Specker’s proof of Infinity in NF does not seem to go through in any useful way) but the consistency problem for INF remains open in spite of the apparent weakness of the theory.