The vertex bound conjecture for critical graphs of tree-depth k

Let $G$ be a graph of tree-depth $k$, meaning that $k$ is the least number of labels in a vertex ranking of $G$ such that every path joining two vertices with the same label contains a vertex with a higher label. A graph is critical if it has tree-depth $k$ and every proper minor has smaller tree-depth. Vertex bound conjecture. Every critical graph with tree-depth $k$ has at most

$$2^{k-1}$$

vertices. This conjecture concerns the possible size of minor-minimal obstructions at each tree-depth; the supplied source does not state whether the bound has been proved or disproved.