Global structure conjecture for the stable manifold of plane-wave equilibria

The dynamical system has plane-wave fixed points $\mathrm{PW}^{\pm}$, a fixed point source $\mathrm{RT}$, a fixed point $\mathrm{PW}^{0}$, a fixed point $\mathrm{T}_3$, and variables $R_3$ and $N_-$. Let $\mathcal{W}^s(\mathrm{PW}^{\pm})$ denote the stable manifold of $\mathrm{PW}^{\pm}$, and let $\mathcal{HO}$ be the invariant subset containing the orbit from $\mathrm{RT}$ to $\mathrm{PW}^{0}$. The system has the discrete symmetry $(R_3,N_-)\mapsto -(R_3,N_-)$. Global stable-manifold conjecture. (i) Every orbit in $\mathcal{W}^s(\mathrm{PW}^{\pm})$ originates from $\mathrm{RT}$: its $\alpha$-limit set is $\mathrm{RT}$. The orbit $\mathrm{RT}\rightarrow\mathrm{PW}^{0}$ in $\mathcal{HO}$ divides $\mathcal{W}^s(\mathrm{PW}^{\pm})$ into the two parts $\mathrm{RT}\rightarrow\mathrm{PW}^{+}$ and $\mathrm{RT}\rightarrow\mathrm{PW}^{-}$. (ii) The boundaries of these two parts, and hence of the whole stable manifold, are two heteroclinic orbits $\mathrm{RT}\rightarrow\mathrm{T}_3$: one approaches $\mathrm{T}_3$ from the positive $(R_3,N_-)$ direction and the other from the negative $(R_3,N_-)$ direction, related by the stated discrete symmetry. This conjecture concerns the global extension and boundary characterization of the locally foliated center-stable manifold; its resolution is not supplied in the source.