Asymptotic gap for treedepth obstructions

Close the asymptotic gap between the known upper and lower bounds on the maximum size of minimal obstructions for graphs of treedepth at most $d$.

Background

For every fixed treedepth bound, the class of graphs of treedepth at most $d$ has finitely many minimal forbidden induced subgraphs. The survey states that these obstructions have an upper bound of $d^{O(d)}$ vertices, while examples with $2^d$ vertices exist, leaving a substantial asymptotic gap.