Schröder enumeration conjecture for the $\underline{23}1$ sorting class

Let $\text{SC}_{\underline{23}1}$ be the vincular-pattern-avoiding stack-sorting map, let $\text{Sort}_n(\text{SC}_{\underline{23}1})$ denote its sorting class in $\mathfrak S_n$, and let $S_{n-1}$ denote the corresponding Schröder number. Schröder enumeration conjecture. The sorting class is enumerated by

$$|\text{Sort}_n(\text{SC}_{\underline{23}1})|=S_{n-1}.$$

The first ten computed terms agree with OEIS sequence A006318, the Schröder numbers, motivating this enumeration conjecture. The supplied text gives no proof or resolution.