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 $SC_σ$, where the stack must always avoid a given permutation $σ$ consecutively. We address one of the main conjectures raised by Defant and Zheng in their dynamical approach to $SC_σ$. Specifically, we show that the periodic points of $SC_σ$ are precisely the permutations that consecutively avoid $σ$ and its reverse.