Second-largest preimage conjecture for the $1\underline{23}$ stack-sorting map

Let $\mathfrak S_n$ be the set of permutations of length $n$, and let $\text{SC}_{1\underline{23}}$ be the vincular-pattern-avoiding stack-sorting map. Second-largest-preimage conjecture. For $n\ge 3$, the second-largest number of preimages that a permutation in $\mathfrak S_n$ can have under $\text{SC}_{1\underline{23}}$ is

$$2^{n-3}.$$

Moreover, exactly $2n-2$ permutations $\pi\in\mathfrak S_n$ satisfy

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

The maximum preimage count for this map is discussed as $2^{n-2}$; this conjecture concerns the next distinct preimage count and its multiplicity. Its resolution is not given in the supplied text.