Global persistency conjecture for compartmental reaction-diffusion networks

Let $X_0\in\mathbb{R}_+^{mN}$ be an initial condition for the closed compartmental reaction-diffusion system, with trajectories denoted by $t\mapsto X(t)$. Global persistency conjecture. Every trajectory with initial condition $X_0\in\mathbb{R}_+^{mN}$ satisfies

$$\mathrm{lim}\,\mathrm{inf}_{t\rightarrow\infty}X(t)>0.$$

The positive orthant is forward invariant for this system, and the global persistency property is assumed in order to exclude possible boundary equilibria. The parser supplies no resolution evidence for this formulation, so its status remains open.