Francisco–Hà–Van Tuyl's expansion conjecture for critically chromatic graphs

Let $G$ be a graph that is critically $s$-chromatic, meaning that its chromatic number is $s$ and deleting any vertex lowers the chromatic number. For a subset $W$ of the vertices of $G$, let $G[W]$ denote the graph obtained by expanding every vertex of $W$ into two adjacent vertices, each inheriting all the original neighbors. Francisco–Hà–Van Tuyl's expansion conjecture. There exists a subset $W$ of the vertices of $G$ such that $G[W]$ is critically $(s+1)$-chromatic. This conjecture is proposed as a possible approach to proving persistence for cover ideals of imperfect graphs; the source gives affirmative persistence results for odd holes and odd antiholes but does not state a resolution of the expansion conjecture.