Generalized graph realization conjecture for heteroclinic networks

Let $G$ be a directed graph. Theorem~ gives a realization result for graphs without $2$-cycles or $\Delta$-cliques.

Generalized realization conjecture. The conclusion of Theorem~ holds even for directed graphs $G$ that may contain $2$-cycles and $\Delta$-cliques.

The conjecture proposes that the restrictions excluding $2$-cycles and $\Delta$-cliques are unnecessary, extending the construction to the more general class of directed graphs. Other realization methods already give robust realizations under the weaker assumption that $G$ has no $1$-cycles, but this stronger extension is not established here.