The large-inertia and large-initial-velocity oscillator-death conjecture for the inertial Winfree model

Let $\mathcal{V}\in\mathbb{R}^N$ be the vector of intrinsic velocities, and let $\kappa_c(\mathcal{V})$ denote the critical coupling strength of the Winfree system. Let initial data $(\Theta^0,\Omega^0)\in\mathbb{R}^{2N}$ and positive inertia $m>0$ be given. The large-inertia and large-initial-velocity oscillator-death conjecture. If

$$\kappa\geq\kappa_c(\mathcal{V}),$$

then, for any initial data $(\Theta^0,\Omega^0)\in\mathbb{R}^{2N}$ and any positive inertia $m>0$, the solution $\Theta(t)$ to the inertial Winfree system exhibits oscillator death. This is a more ambitious open problem than the preceding uniform-constant conjecture because it imposes no smallness condition on the inertia or initial velocity.