The expansion conjecture for critically chromatic graphs

Let $G=(V,E)$ be a simple graph, where $V$ is its vertex set. A graph is critically $s$-chromatic if it has chromatic number $s$ and deleting any vertex lowers its chromatic number. For a subset $W\subseteq V$, let $G[W]$ denote the graph obtained by expanding successively at the vertices in $W$, where expansion at a vertex $x$ adds a new vertex adjacent to $x$ and to every neighbor of $x$. Expansion conjecture. If $G$ is a critically $s$-chromatic graph, then there exists a subset $W$ of the vertices such that $G[W]$ is critically $(s+1)$-chromatic. This graph-theoretic conjecture was intended to yield the persistence property for cover ideals of graphs. It was disproved by Kaiser, Stehlík and Škrekovski, who exhibited a family of counterexamples; the paper studies these counterexamples further in relation to persistence and non-increasing depth.