The conjecture that every critical graph is 1-unique

A graph $G$ is 1-unique if, for every vertex $v$ of $G$, there is a tree-depth labeling using labels from $\{1,\dots,\operatorname{td}(G)\}$ in which $v$ is the unique vertex receiving label $1$. A graph is critical when it is $k$-critical for its tree-depth $k$: every proper minor has smaller tree-depth.

Every-critical-graph conjecture. Every critical graph is 1-unique.

The conjecture is motivated by the fact that every critical graph with tree-depth at most $4$ is 1-unique, and that critical graphs produced by certain known constructions are also 1-unique. Its status is not resolved by the supplied text.