The density-threshold conjecture for global synchrony

A graph is globally synchronizing when its adjacency matrix has only synchronized global minima of the Kuramoto energy. Density-threshold conjecture. For every $\varepsilon>0$, there exist a positive integer $n$ and a graph $G$ on $n$ vertices whose minimum degree is at least $\left(\frac34-\varepsilon\right)n$, but which is not globally synchronizing. The $\frac34$ threshold is supported by known upper bounds and dynamical evidence, but the conjectured extremal construction remains open.