The universal fertility conjecture for consecutive-pattern-avoiding stack-sorting maps

Let $S_3$ be the set of permutations of length $3$, and let $\operatorname{SC}_\sigma$ be the consecutive-pattern-avoiding stack-sorting map associated with $\sigma$.

Universal fertility conjecture. For every $\sigma\in S_3$ and every positive integer $f$, there exists a permutation $\pi$ such that

$$|\operatorname{SC}_\sigma^{-1}(\pi)|=f.$$

This contrasts with the classical stack-sorting map, which has known infertility numbers. The assertion remains open for the consecutive-pattern-avoiding maps.