The conjecture on globally synchronizing random 3-regular graphs

Let a graph be globally synchronizing when its adjacency matrix is globally synchronizing, meaning that the only local minima of the Kuramoto energy are the global synchronized minima. Global synchronization conjecture. A uniformly random $3$-regular graph is globally synchronizing with high probability, meaning with probability tending to $1$ as $n\to\infty$. Uniform random $d$-regular graphs are known to be globally synchronizing with high probability for $d\geq600$, whereas the $3$-regular case remains open.