4 problems

$O(G)$ conjecture. These equivalent statements hold when $\mathcal{F}$ is the class of all strongly connected graphs.

$O(G)$ conjecture. $O(G)$ is well-defined for every strongly connected graph, although the conjecture does not immediately provide a method for computing it.

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

Bunchy factor conjecture. The barrier to proving the $O(G)$ conjecture is our lack of a sufficiently general method of producing homomorphisms with nontrivial stability relation.