Universal pathwise critical coupling conjecture for the Winfree model

Let $\Omega=\\{\omega_1,\cdots,\omega_N\\}\in \mathbb{R}^N$ be a frequency vector, and let $\Theta^0=\\{\theta_1^0,\cdots,\theta_N^0\\}$ be an initial phase vector. Define the critical coupling strength $\kappa_{\mathrm{c}}(\Omega)$ and pathwise critical coupling strength $\kappa_{\mathrm{pc}}(\Theta^0,\Omega)$ as the infima of the coupling thresholds above which the Winfree system admits an equilibrium and the solution from $\Theta^0$ converges, respectively. Universal pathwise coupling conjecture. There exists a universal constant $c>0$ such that

$$\kappa_{\mathrm{pc}}(\Theta^0,\Omega)\le c\max_{i=1,\cdots,N}|\omega_i|,\quad \forall \Theta^0=\\{\theta^0_1,\cdots,\theta^0_N\\}\in \mathbb{R}^N,~\forall \Omega=\\{\omega_1,\cdots,\omega_N\\}\in \mathbb{R}^N.$$

The theorem cited in the source establishes the analogous bound for almost every initial phase vector, while this conjecture asks for a uniform bound over every initial phase vector and frequency vector; the source gives no resolution.