Nonexistence of infinite uniquely-Wilf classes containing all permutations of length three

Let ${\mathcal{C}}$ be a class of permutations, and let ${\mathcal{C}}_3$ denote its permutations of length three. Let ${\mathcal{S}}_3$ be the set of all permutations of length three. Nonexistence conjecture. There are no infinite uniquely-Wilf classes ${\mathcal{C}}$ with

$${\mathcal{C}}_3={\mathcal{S}}_3.$$

This would complete the classification of infinite uniquely-Wilf classes in the remaining cases where ${\mathcal{C}}_3={\mathcal{S}}_3$ and the number of permutations of length four is $22$, $23$, or $24$; the paper explains that these cases remain open because the available computational constraints were insufficient.