Smallness of the class $\mathcal{Q}$

Let $\mathcal{Q}$ be the graph class defined in the preceding construction. A graph class is small if there is a constant $c$ such that the number of its labelled $n$-vertex graphs is at most $c^n n!$. Smallness conjecture for $\mathcal{Q}$. There exists a constant $c$ such that the number of $n$-vertex graphs in $\mathcal{Q}$ is at most

$$c^n n!,$$

i.e. the class $\mathcal{Q}$ is small. If true, this would make $\mathcal{Q}$ an explicit counterexample to the small conjecture, which asserts that every small class has bounded twin-width; the supplied text does not resolve the conjecture.