Small hereditary classes and bounded merge-width

Prove that every small hereditary graph class has bounded merge-width.

Background

A small hereditary class contains at most n!⋅2O(n)n!\cdot 2^{O(n)} labelled graphs on nn vertices. The proposed conjecture would extend known results for weakly sparse small hereditary classes and small hereditary classes of ordered graphs, and would strengthen existing connections between smallness and monadic dependence.

References

On the other hand, we conjecture that if a hereditary graph class is small -- contains at most n!\cdot 2{O(n)} distinct labeled n-vertex graphs -- then it has bounded merge-width.

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section “Discussion”, paragraph “Combinatorial properties”