Conjecture on coincidence of pathwise and phase-locking critical coupling strengths

Let $u_1,\ldots, u_N$ be fixed natural frequencies, and let $u=( u_1,\ldots, u_N)$. Consider the inertial Kuramoto system $$, with coupling strength $\kappa$, phase initial data $\Theta^0$, and velocity initial data $\Omega^0$. Asymptotic phase-locking means that the phase differences converge to limiting values and the velocities converge to a common limiting frequency. The pathwise critical coupling strength and critical coupling strength coincide. There \exists a finite value $\kappa_c=\kappa_c(\nu_1,\ldots,\nu_N)$ such that: (1) if $\kappa\geq\kappa_c$, then for Lebesgue almost every initial data $(\Theta^0,\Omega^0)$, system $$ exhibits asymptotic phase-locking; and (2) if $\kappa<\kappa_c$, then system $$ has no phase-locked states. This conjecture formalizes the expected generic synchronization threshold for the inertial Kuramoto model; the supplied text gives numerical motivation but no resolution.