Smallness characterisation conjecture for graph classes by twin-width

Let $\mathcal{C}$ be a class of graphs closed under induced subgraphs. Call $\mathcal{C}$ small if there is a constant $c$ such that, for every $n$, it contains at most $n!\,c^n$ graphs on vertex set $\{1,\dots,n\}$. Smallness characterisation conjecture. The class $\mathcal{C}$ is small if and only if it has bounded twin-width. This would characterise bounded twin-width among hereditary graph classes and extend the corresponding growth dichotomy for matrices; the source gives no resolution of the conjecture.