Equality of the sorting classes for the 312-type stack-sorting maps

Let $\text{SC}_{312}$, $\text{SC}_{\underline{31}2}$, and $\text{SC}_{3\underline{12}}$ be the vincular-pattern-avoiding stack-sorting maps, and let $\text{Sort}_n(f)$ denote the permutations in $\mathfrak S_n$ mapped to the identity by $s\circ f$. Equality conjecture. The sorting classes of $\text{SC}_{\underline{31}2}$ and $\text{SC}_{3\underline{12}}$ are identical:

$$\text{Sort}_n(\text{SC}_{312})=\text{Sort}_n(\text{SC}_{\underline{31}2})=\text{Sort}_n(\text{SC}_{3\underline{12}}).$$

Furthermore, for every $\tau\in\text{Sort}_n(\text{SC}_{312})$,

$$\text{SC}_{312}(\tau)=\text{SC}_{\underline{31}2}(\tau)=\text{SC}_{3\underline{12}}(\tau).$$

The maps $\text{SC}_{\underline{31}2}$ and $\text{SC}_{3\underline{12}}$ are being compared because their sorting-class cardinalities agree in the computed cases and the corresponding classical and vincular maps coincide in related free-stack situations. The equality of the classes and of their values on the sorting class remains conjectural.