Bi-stability conjecture for the fluid model

Let $x$ be a fluid solution with initial condition $x(0)$, let ${\cal O}$ be the set of initial conditions whose trajectories oscillate indefinitely, and let $u^*$ denote a periodic equilibrium when it exists. Let ${\cal S}_{u^*}$ be the stability set of $u^*$, and let $\mathbb{S}$ be the state space. Bi-stability conjecture. If $x(0) \in {\cal O}$, then there exists a unique periodic equilibrium $u^*$ and $x$ converges to $u^*$ as in the paper's definition of convergence. Consequently, ${\cal S}_{x_0^*} \cup {\cal S}_{u^*}=\mathbb{S}$, so the fluid model is bi-stable and every fluid trajectory converges to one of the two equilibria as $t\to\infty$. The conjecture asserts that all indefinitely oscillating fluid trajectories converge to the unique periodic equilibrium; together with the proved convergence of non-oscillating trajectories to the unique stationary point, this would establish global bi-stability of the fluid limit. Further context on what is known beyond these results is not provided in the source.