The 1-uniqueness conjecture for critical graphs

Let $G$ be a critical graph, meaning that it has tree-depth $k$ for some $k$ and every proper minor of $G$ has smaller tree-depth. A graph is 1-unique if, for every vertex $v$ of $G$, some optimal ranking of $G$ has $v$ as the unique vertex with label $1$. The 1-uniqueness conjecture. Every critical graph is 1-unique. The source says that this conjecture implies the maximum-degree conjecture and gives no resolution of it; the paper studies classes satisfying it.