The R_0-independent pathwise oscillator-death conjecture for the inertial Winfree model

Let $N$ oscillators have initial data $(\{\theta_i^0\}_{i=1}^N,\{\omega_i^0\}_{i=1}^N)$ and system parameters $(\{\nu_i\}_{i=1}^N,\kappa,m)$. Write $\mathcal{V}=(\nu_i)_{i=1}^N$ and let $\Omega^0=(\omega_i^0)_{i=1}^N$. The R_0-independent pathwise oscillator-death conjecture. There exist absolute constants $a,b,c>0$ with the following property. For any initial data and system parameters satisfying

$$\frac{\|\mathcal{V}\|_\infty}{\kappa}<a,\qquad m\kappa<b,\qquad \frac{\|\Omega^0\|_\infty}{\kappa}<c,$$

the solution $\Theta$ to the inertial Winfree system exhibits oscillator death: for every $i\in[N]$, the limits $\theta_i^\infty:=\lim_{t\to\infty}\theta_i(t)$ and $\lim_{t\to\infty}\dot\theta_i(t)=0$ exist. This conjecture seeks constants independent of the initial-data scale $R_0$, improving the explicit but nonuniform $R_0^{3/2}$ dependence in the proved pathwise theorem.