Conjecture on generic-data asymptotic phase-locking in the inertial Kuramoto model

Let $R(t)$ be the order parameter, let $\mathcal{D}(\mathcal{V})$ and $\operatorname{Var}(\mathcal{V})$ be the dispersion and variance of the natural-frequency data, let $\kappa$ be the coupling strength, and let $\Theta^0$ be initial phase data. The phrase generic initial data has the meaning used in the paper.

Generic-data phase-locking conjecture. (Weak form) For every $\varepsilon>0$, there exists $\delta=\delta(\varepsilon)$ such that, if $\mathcal{D}(\mathcal{V})/\kappa<\delta$, then for generic initial data $\Theta^0$,

$$\liminf_{t\to\infty}R(t)\ge 1-\varepsilon.$$

(Strong form) For every $\varepsilon\in(0,\frac12)$, there exists a sufficiently small constant $c_\varepsilon>0$ such that, if $\operatorname{Var}(\mathcal{V})/\kappa<c_\varepsilon$, then for generic initial data $\Theta^0$,

$$1-\left(\frac12+\varepsilon\right)\frac{\operatorname{Var}(\mathcal{V})}{\kappa^2}\le\liminf_{t\to\infty}R(t)\le\limsup_{t\to\infty}R(t)\le 1-\left(\frac12-\varepsilon\right)\frac{\operatorname{Var}(\mathcal{V})}{\kappa^2}.$$

If, in addition, $\mathcal{D}(\mathcal{V})/\kappa<c_\varepsilon$, then $\Theta(t)$ converges to the unique phase-locked state of confined in the quarter circle.

The weak form predicts near-complete asymptotic synchronization for generic data when frequency dispersion is small relative to coupling. The strong form gives quantitative bounds and convergence to a specified phase-locked state; the source presents these claims as conjectural.