The maximum-degree conjecture for critical graphs of tree-depth

Let $G$ be a critical graph with tree-depth $k$, meaning that $G$ has tree-depth $k$ and every proper minor of $G$ has smaller tree-depth. The maximum-degree conjecture. The maximum degree of $G$ is at most

$$k-1.$$

The paper presents this as one of two conjectures motivating its construction of critical and 1-unique graph classes; no resolution is given in the supplied text.