Global convergence conjecture for the dynamic Kuramoto model

Let the dynamic Kuramoto model be given by with zero natural frequencies, and let the phase vector start from a deterministic point outside the set of spurious fixed points $\mathcal{S}$. The phase space is the torus $\mathbb{T}^N$, and convergence is understood up to a global translation. Global convergence conjecture. For every $\epsilon\in(0,\infty)$ and every deterministic choice of $\pmb\theta^\epsilon(0)\in\mathbb{T}^N\backslash\mathcal{S}$, almost surely,

$$\lim_{t\to\infty}\pmb\theta^\epsilon(t)=\mathbf 0.$$

The analogous assertion is expected for the corresponding static random conductance model after the appropriate change of notation. It says that the synchronized state attracts every initial condition outside the spurious fixed-point set, while the surrounding discussion indicates that the absence of stable manifolds near those fixed points remains unproved.