Global phase-locking conjecture for homogeneous Kuramoto oscillators

Let $N$ phase oscillators evolve according to the homogeneous Kuramoto system

$$\dot{\theta}(t)=k f(\theta(t)),\qquad \theta(t_0)=\theta_0,$$

where $\theta_0\in\mathbb{R}^N$ and $R(\theta)$ is the magnitude of the order parameter. Global phase-locking conjecture. For almost all initial conditions $\theta_0$, the solution satisfies

$$\lim_{t\rightarrow\infty}R(\theta(t))=1.$$

The claim predicts global phase-locking for almost every initial condition, although other invariant manifolds prevent global asymptotic stability in the usual sense; the source notes that the phase-locking manifold is locally asymptotically stable.