Conjecture on almost-everywhere limit sets of differentially positive systems

Let $\Sigma$ be a continuous dynamical system on a vector space $\mathcal{X}$ of the form

$$\dot{x}=f(x),$$

and let $\mathcal{K}_{\mathcal{X}}$ be the cone field used in Theorem

. Assume the hypotheses of that theorem, and write $\omega(\xi)$ for the $\omega$-limit set of the trajectory starting at $\xi$. **Limit-set conjecture.** Under the assumptions of Theorem

, for almost every $\xi\in\mathcal{X}$, the $\omega$-limit set $\omega(\xi)$ is given by either a fixed point, a limit cycle, or fixed points and connecting arcs.

This conjecture proposes that, despite the potentially complex limit sets of general differentially positive systems, almost every trajectory has an $\omega$-limit set with one of these simple forms. The supplied context does not state whether the conjecture has been proved or disproved.