Unique maximizers 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. Unique-maximizers conjecture. The maximum of $|\text{SC}_{1\underline{23}}^{-1}|$ is uniquely achieved by

$$\pi=(n-1)(n-2)\cdots 1n,$$

and the maximum of $|\text{SC}_{3\underline{21}}^{-1}|$ is uniquely achieved by

$$\pi=23\cdots n1.$$

The claim refines the asserted maximum preimage count $2^{n-2}$ by specifying its unique attaining permutation for each map. The supplied text presents this as a conjecture following partial smoothing results; no resolution is given.