Conjecture on weak Lyapunov functionals for the inertial and first-order Kuramoto models

Let $\mathcal{V}$ denote the natural-frequency data, let $\mathcal{D}(\mathcal{V})$ and $\operatorname{Var}(\mathcal{V})$ denote the dispersion and variance quantities defined in the paper, and let $\kappa$ be the coupling strength. A weak Lyapunov functional is a function $f:\mathbb{T}\times\mathbb{R}^N\to\mathbb{R}$ for, or a function $g:\mathbb{T}^N\to\mathbb{R}$ for, whose derivative along every solution is nonnegative, with equality only at phase-locked states traveling at constant speed and at the configurations described in Example.

Weak-Lyapunov-functional conjecture. Either there is a constant $c\ge\frac12$ such that, whenever $\kappa>c\mathcal{D}(\mathcal{V})$, both models admit weak Lyapunov functionals, or, writing $\kappa_c(\mathcal{V})$ for the critical coupling strength above which phase-locked states exist, whenever $\kappa>\kappa_c(\mathcal{V})$, both models admit weak Lyapunov functionals.

The conjecture asks for monotone quantities that would identify the relevant asymptotic states and provide a Lyapunov-based explanation of synchronization in the large-coupling regime. The source gives no resolution.