Maximum preimage conjecture for the $1\underline{23}$ and $3\underline{21}$ stack-sorting maps

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ and $\text{SC}_{3\underline{21}}$ be the corresponding vincular-pattern-avoiding stack-sorting maps. Maximum-preimage conjecture. For every $n\ge 2$,

$$\max_{\pi\in\mathfrak S_n}|\text{SC}_{1\underline{23}}^{-1}(\pi)|=\max_{\pi\in\mathfrak S_n}|\text{SC}_{3\underline{21}}^{-1}(\pi)|=2^{n-2}.$$

The claim gives the maximum possible preimage count for both maps. The surrounding paper develops bounds and related constructions, but the supplied text does not state a resolution of this assertion.