Uniform boundedness conjecture for the pathwise critical coupling strength

Let $N$ be the number of particles, let $\Theta^0$ be an initial phase vector, and let $\Omega=(\nu_1,\ldots,\nu_N)$ be the vector of natural frequencies. Write $D(\Omega)$ for the diameter of the frequencies and $\kappa_{pc}(\Theta^0,\Omega,N)$ for the pathwise critical coupling strength. The conjecture concerns almost every choice of $\Theta^0$ and $\Omega$. Uniform boundedness conjecture. The pathwise critical coupling strength is uniformly bounded over almost every $\Theta^0$ in either of the following senses: there is a universal constant $C\geq 1$, independent of $N$, $\Theta^0$, and $\Omega$, such that

$$\kappa_{pc}(\Theta^0,\Omega,N)\leq C D(\Omega)$$

for almost every $\Theta^0$ and $\Omega$; and, in the strong version, one may take $C=1$, so that

$$\kappa_{pc}(\Theta^0,\Omega,N)\leq D(\Omega).$$

The weak and strong versions are motivated by numerical simulations, while the pathwise critical coupling strength is otherwise poorly understood. The source gives no resolution of either version.