Uniform boundedness of Goldbach graph diameters

Prove that the diameters of all finite Goldbach graphs and all finite near Goldbach graphs are bounded by a universal constant independent of the number of vertices.

Background

The paper reports computational evidence that finite Goldbach graphs and near Goldbach graphs have diameter at most 5 for graphs with up to 10,000 vertices. The authors formulate a conjecture that this bounded-diameter behavior persists uniformly for all finite graphs in both families. The proven results only establish diameter at most 3 for individual prime multiple missing graphs and at most 4 for the intersection $G(3,n)\cap G(5,n)$, leaving the general uniform bound unresolved.