Transitive containment in flat replacement and flat collection theories

Determine whether the theories fReR^{\neg \infty} and \mathsf{fReC}^{\neg \infty}, obtained by adding the negation of Infinity to the flat replacement and flat collection weakenings of \mathsf{ReR}, respectively, prove the Axiom of Transitive Containment \mathsf{TCo}.

Background

The paper defines \mathsf{fReR} by replacing \Delta_0\textsf{-Replacement} in \mathsf{ReR} with Flat \Delta_0\textsf{-Replacement}, and defines \mathsf{fReC} by replacing it with Flat \Delta_0\textsf{-Collection} together with \Delta_0\textsf{-Separation}. Their corresponding theories with superscript \neg\infty additionally include the negation of the Axiom of Infinity. The paper establishes that \mathsf{ReR} proves transitive containment and constructs a model showing that the analogous result can fail for a related theory with collection, but it does not resolve whether either of these two finite theories proves \mathsf{TCo}.