Descending sequences in reflection hierarchies

There is no recursively enumerable sequence of sufficiently strong 2-consistent r.e. theories such that each proves the $2$-consistency of the next. Montalbán and Shavrukov independently asked whether this result generalizes to $0^{\mathrm{\prime }}$-recursive sequences. We consider a general version of this problem: For arbitrary $n$, for which complexity classes $\mathrm{\Gamma }$ are there $\mathrm{\Gamma }$-definable sequences of $n$-consistent r.e. theories each of which proves the $n$-consistency of the next? The answer to this question depends not only on $n$ and $\mathrm{\Gamma }$ but also on the manner in which sequences are encoded in arithmetic. We provide positive answers for certain encodings and negative answers for others.