The compatibility-count conjecture for $(3,n)$ generalized Feynman diagrams

Let a $(3,n)$ generalized Feynman diagram (GFD) be a combinatorial object compatible with color orderings in the sense of the paper. Compatibility-count conjecture. Every $(3,n)$ GFD is compatible with exactly

$$2^{2(n-4)}$$

color orderings. This is motivated by the complete enumeration through $(3,8)$, where every GFD has the predicted number of compatible generalized color orderings; the assertion is proposed for general $n$.