Totality of the inverse Ackermann interpretation

Determine whether the theories \mathsf{fReR}^{\neg \infty} and \mathsf{fReC}^{\neg \infty} prove the totality of the inverse Ackermann interpretation, namely, whether every set is assigned a natural number by that interpretation.

Background

The paper proves that \mathsf{ReR}^{\neg\infty} defines a bijection between sets and natural numbers by establishing the totality and bijectivity of the inverse Ackermann interpretation. It then compares this result with weaker theories involving flat replacement and flat collection. Whether those theories retain enough strength to prove totality of the inverse Ackermann interpretation is left unresolved.