Blowup recolorability conjecture for prime graphs

Let $G$ be a prime graph, meaning that $G$ has no nontrivial module. A blowup of $G$ is obtained by replacing each vertex of $G$ with a nonempty independent set and each edge with all edges between the corresponding sets. A graph is recolorable if its recoloring graph is connected.

Blowup recolorability conjecture. Every blowup of $G$ is recolorable if and only if every induced subgraph of $G$ is recolorable.

This conjecture would characterize when all blowups of a prime graph are recolorable in terms of the recolorability of its induced subgraphs. The paper notes that the analogous reduction to blowups is relevant to the question of whether recolorability of all prime graphs in a graph class implies recolorability of every graph in the class, and gives examples showing that a recolorable prime graph can have a non-recolorable blowup.