The $O(G)$ conjecture for strongly connected graphs

Let $G$ be a strongly connected graph. A graph $H$ is below $G$ in the relation $40_S$ when there is a synchronizing right resolver from $G$ to $H$; write $H\leq_S G$.

$O(G)$ conjecture. For every strongly connected graph $G$, there is a unique $\leq_S$-minimal graph $O(G)$ satisfying

$$O(G)\leq_S G.$$

This conjecture asserts the existence and uniqueness of a canonical minimal synchronizing factor for each strongly connected graph. The supplied text gives no resolution status.