The $O(G)$ conjecture on minimal synchronizing factors

Let $G$ be a strongly connected graph. Write $H\fold_S G$ when $H$ is a synchronizing factor of $G$, and let $O(G)$ denote a candidate minimal synchronizing factor.

$O(G)$ conjecture. The set of graphs $H$ with $H\fold_S G$ has a unique $\fold_S$-minimal element $O(G)$.

This conjecture asserts that every strongly connected graph has a well-defined canonical minimal synchronizing factor. It is known in several classes, including graphs related to the road-colouring theorem and almost bunchy graphs, but is open in general.