Global dynamics conjecture for the Langford vector field

Let $\alpha$ be the parameter of the Langford vector field, and let $\alpha_1\approx0.697144898322973$, $\alpha_2\approx0.823$, $\alpha_3\approx0.925$, $\alpha_4\approx0.9321697517861$, and $\alpha_5\approx1$ be the successive bifurcation values. Denote by $\gamma$ the periodic orbit before the Neimark–Sacker bifurcation, by $\mathcal{T}$ the invariant torus, by $\gamma_1$ and $\gamma_2$ the periodic orbits created at the resonance, by $p_0$, $p_1$, and $p_2$ the relevant equilibria, by $W^u(p_0)$ and $W^s(p_1)$ their unstable and stable manifolds, and by $\widetilde{\mathcal{T}}$ the chaotic attractor formed after the breakup of the invariant torus. Global dynamics conjecture. The flow generated by the Langford vector field has the following properties: for $0<\alpha<\alpha_1$, $\gamma$ is the global attractor; for $\alpha_1<\alpha<\alpha_2$, the global attractor is either $\mathcal{T}$ or $\mathcal{T}\cup\gamma_1$; for $\alpha_2<\alpha<\alpha_3$, the global attractor is $\mathcal{T}$; for $\alpha_3<\alpha<\alpha_4$, the global attractor is $\mathcal{T}\cup p_2$; for $\alpha_4<\alpha<\alpha_5$, there is multistability and the global attractor contains at least $\widetilde{\mathcal{T}}$, the attracting periodic orbit $\gamma_2$, and the attracting equilibrium $p_2$; for $\alpha>\alpha_5$, $p_2$ is the global attractor; for $0<\alpha<\alpha_3$, $W^u(p_0)$ accumulates on the global attractor, which is either $\gamma$ until $\alpha=\alpha_1$ or $\mathcal{T}$ thereafter; and for $\alpha_3<\alpha<\alpha_5$, $W^s(p_1)$ is a separatrix, with the basin of attraction of $p_2$ outside the bubble formed by $W^s(p_1)$. These claims summarize the numerically suggested global dynamics between the listed bifurcation values; their status is not established by the supplied text and should therefore be regarded as open.