Conjecture on the omega-limit set for an extended dissipative system

Let $u: \mathbb{R} \times \mathbb{R}_+ \to \mathbb{R}$ solve the parabolic equation with initial data $u_0$ satisfying

$$u_0(x)=(-1)^{n+1}\quad\text{if }b_n\leq |x|<b_{n+1},$$

where $(b_n)_{n\in\mathbb{N}}$ is strictly increasing, $b_0=0$, and $b_{n+1}\gg b_n$ for all $n\in\mathbb{N}$. Let $\mathcal{T}_{\mathrm{loc}}(\mathbb{R})$ be the topology of uniform convergence on compact subsets of $\mathbb{R}$, and let $u_\psi$ be the eternal solution of characterized by

$$\sup_{x\in\mathbb{R}}\left|u_\psi(x,t)-V_{a(t)}(x)\right|\xrightarrow[t\to-\infty]{}0,$$

with $a(t)\to+\infty$ as $t\to-\infty$. Write $u_\pm=\pm1$. Omega-limit set conjecture. The omega-limit set of the trajectory $(u(t))_{t\geq0}$ in $\mathcal{T}_{\mathrm{loc}}(\mathbb{R})$ is

$$\omega=\{u_+,u_-\}\cup\{u_\psi(t)\mid t\in\mathbb{R}\}\cup\{-u_\psi(t)\mid t\in\mathbb{R}\}.$$

The conjecture describes the coarsening dynamics generated by infinitely many widely separated kink pairs: successive annihilations are expected to produce not only the two constant equilibria but also the eternal heteroclinic trajectories and their negatives as local omega-limits. The supplied text gives the heuristic motivation but no resolution, so the conjecture remains open.