  West's stack-sorting map involves a stack which avoids the permutation $21$
consecutively. Defant and Zheng extended this to a consecutive-pattern-avoiding
stack-sorting map $SC_\sigma$, where the stack must always avoid a given
permutation $\sigma$ consecutively. We address one of the main conjectures
raised by Defant and Zheng in their dynamical approach to $SC_\sigma$.
Specifically, we show that the periodic points of $SC_\sigma$ are precisely the
permutations that consecutively avoid $\sigma$ and its reverse.
