The small conjecture for hereditary graph classes

Let a graph class be small if the number of its labeled graphs on vertex set $[n]$ is at most $n!c^n$ for some constant $c$, and let a graph class be hereditary if it is closed under taking induced subgraphs. A class has bounded twin-width if there is an integer $d$ such that every graph in the class has twin-width at most $d$.

Small conjecture. Every small hereditary class has bounded twin-width.

The conjecture is presented as a converse to the theorem that every class with bounded twin-width is small. The paper does not answer the question, but gives evidence by proving bounded twin-width for many potential counterexamples.